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Fj(\()p Fh(j)275 b Fg(\241)200 b Fj(1\))387 b(for)h(the)e(second,)397 b Fh(j)75 b Fj(\()p Fh(j)274 b Fg(\241)200 b Fj(1\))p Fh(=)p Fj(2)389 b(for)e(the)g(third,)395 b(and)387 b Fh(j)75 b Fj(\()p Fh(j)275 b Fg(\241)200 b Fj(1\)\()p Fh(j)275 b Fg(\241)200 b Fj(2\))p Fh(=)p Fj(6)388 b(for)g(the)e(fourth.)563 b(Adding)800 33718 y(these)433 b(terms)g(and)g(simplifying)j(sho)-36 b(ws)433 b(that)17880 36137 y Fh(C)18811 36336 y Fl(j)19297 36137 y Fj(\()p Fh(j)370 b Fj(+)295 b(6)p Fh(;)23250 35853 y Fj(^)23250 36137 y(2)q(\))369 b(=)f Fh(j)75 b Fj(\()p Fh(j)27888 35588 y Fc(2)28709 36137 y Fj(+)295 b(6)p Fh(j)370 b Fg(\241)295 b Fj(1\))p Fh(=)p Fj(6)p Fh(:)800 38555 y Fj(The)454 b(sequence)f Fh(C)9835 38754 y Fl(j)10322 38555 y Fj(\()p Fh(j)383 b Fj(+)308 b(6)p Fh(;)14301 38271 y Fj(^)14301 38555 y(2)r(\))453 b(app)36 b(ears)454 b(as)g(A008778)h(in)f(Sloane's)g(On-Line)e(Encyclop)36 b(edia)454 b(of)g(In)-36 b(teger)800 40160 y(Sequences)433 b([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.10) [[137 356 149 368] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)579 b(A008778)435 b(is)f(de\257ned)e(b)-36 b(y)18222 42579 y Fh(a)p Fj(\()p Fh(n)p Fj(\))369 b(=)f(\()p Fh(n)296 b Fj(+)e(1\)\()p Fh(n)27764 42030 y Fc(2)28586 42579 y Fj(+)h(8)p Fh(n)h Fj(+)e(6\))p Fh(=)p Fj(6)800 44997 y(and)490 b(coun)-36 b(ts)490 b(the)g Fh(n)p Fj(-dimensional)h (partitions)g(of)g(4)g(\(for)g(a)g(de\257nition)f(see)h([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.1) [[423 312 429 324] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)506 b(p.)749 b(179\).)h(W)-108 b(e)491 b(need)f(to)800 46603 y(sho)-36 b(w)485 b(the)f(equiv)-72 b(alence)485 b(of)h(the)e(sequences)g Fh(a)p Fj(\()p Fh(n)p Fj(\))h(and)f Fh(C)29525 46802 y Fl(j)30011 46603 y Fj(\()p Fh(j)405 b Fj(+)329 b(6)p Fh(;)34033 46318 y Fj(^)34033 46603 y(2)r(\))484 b(for)h(a)g(suitable)g(v)-72 b(alue)485 b(of)g Fh(n)p Fj(.)732 b(The)800 48208 y(form)-36 b(ulas)358 b(for)h(these)e(t)-36 b(w)g(o)358 b(sequences)g(suggest)g(that)f Fh(C)27546 48407 y Fl(j)28032 48208 y Fj(\()p Fh(j)215 b Fj(+)140 b(6)p Fh(;)31675 47924 y Fj(^)31675 48208 y(2)q(\))369 b(=)g Fh(a)p Fj(\()p Fh(j)215 b Fg(\241)140 b Fj(1\),)373 b(whic)-36 b(h)358 b(can)g(b)36 b(e)357 b(con\257rmed)800 49813 y(b)-36 b(y)434 b(basic)f(algebraic)i (manipulations.)p 52170 49813 V Black 800 52268 a Ff(Theorem)499 b(8)p Black 651 w Fi(The)465 b(eighth)f(diagonal)g(c)-66 b(ontains)464 b(the)g(p)-66 b(entagonal)464 b(pyr)-66 b(amidal)464 b(numb)-66 b(ers.)800 54724 y Ff(Pro)42 b(of.)663 b Fj(F)-108 b(or)431 b Fh(k)414 b Fj(=)369 b(8,)433 b(w)-36 b(e)431 b(need)g(to)h(distribute)e Fh(n)292 b Fg(\241)f Fh(j)444 b Fj(=)368 b(7.)578 b(The)432 b(partitions)f(of)i (7)e(that)g(do)h(not)f(con)-36 b(tain)800 56329 y(a)581 b(1)g(are)g Fg(f)p Fj(7)p Fg(g)p Fj(,)618 b Fg(f)p Fj(5)p Fh(;)221 b Fj(2)p Fg(g)p Fj(,)619 b Fg(f)p Fj(4)p Fh(;)221 b Fj(3)p Fg(g)582 b Fj(and)e Fg(f)p Fj(3)p Fh(;)221 b Fj(2)p Fh(;)g Fj(2)p Fg(g)p Fj(.)1022 b(These)581 b(corresp)36 b(ond,)617 b(resp)36 b(ectiv)-36 b(ely)-108 b(,)618 b(to)581 b(the)f(follo)-36 b(wing)800 57934 y(n)g(um)g(b)36 b(er)476 b(of)i(comp)36 b(ositions:)668 b Fh(j)r(;)699 b(j)75 b Fj(\()p Fh(j)400 b Fg(\241)325 b Fj(1\))p Fh(;)700 b(j)75 b Fj(\()p Fh(j)399 b Fg(\241)326 b Fj(1\),)489 b(and)477 b Fh(j)75 b Fj(\()p Fh(j)399 b Fg(\241)326 b Fj(1\)\()p Fh(j)400 b Fg(\241)325 b Fj(2\))p Fh(=)p Fj(2.)711 b(Adding)477 b(these)g(terms)800 59539 y(and)433 b(simplifying)i(sho)-36 b(ws)434 b(that)19323 61144 y Fh(C)20254 61343 y Fl(j)20740 61144 y Fj(\()p Fh(j)370 b Fj(+)295 b(7)p Fh(;)24693 60860 y Fj(^)24693 61144 y(2)q(\))369 b(=)g Fh(j)28213 60596 y Fc(2)28738 61144 y Fj(\()p Fh(j)h Fj(+)295 b(1\))p Fh(=)p Fj(2)p Fh(:)800 63248 y Fj(The)454 b(sequence)f Fh(C)9835 63447 y Fl(j)10322 63248 y Fj(\()p Fh(j)383 b Fj(+)308 b(7)p Fh(;)14301 62964 y Fj(^)14301 63248 y(2)r(\))453 b(app)36 b(ears)454 b(as)g(A002411)h(in)f(Sloane's)g(On-Line)e(Encyclop)36 b(edia)454 b(of)g(In)-36 b(teger)800 64854 y(Sequences)433 b([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.10) [[137 133 149 145] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)579 b(A002411)435 b(is)f(de\257ned)e(b)-36 b(y)21368 67272 y Fh(a)p Fj(\()p Fh(n)p Fj(\))369 b(=)g Fh(n)26365 66724 y Fc(2)26891 67272 y Fj(\()p Fh(n)295 b Fj(+)g(1\))p Fh(=)p Fj(2)800 69691 y(and)344 b(coun)-36 b(ts)343 b(the)h(p)36 b(en)-36 b(tagonal)345 b(p)-36 b(yramidal)344 b(n)-36 b(um)g(b)36 b(ers)343 b(\(for)i(a)f (de\257nition)g(see)g([)p 0 1 0 0 TeXcolorcmyk(3)p (#cite.3) [[420 90 425 102] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)363 b(pp.)548 b(193-195\).)h(Clearly)-108 b(,)800 71296 y Fh(C)1731 71495 y Fl(j)2218 71296 y Fj(\()p Fh(j)369 b Fj(+)295 b(7)p Fh(;)6170 71012 y Fj(^)6170 71296 y(2)r(\))368 b(=)h Fh(a)p Fj(\()p Fh(j)75 b Fj(\).)p 52170 71296 V Black 26475 74617 a(8)p Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Ff(Theorem)499 b(9)p Black 651 w Fi(The)465 b(ninth)e(diagonal)i(c) -66 b(ontains)463 b(the)i Fh(n)p Fi(-dimensional)f(p)-66 b(artitions)463 b(of)i(5.)800 3845 y Ff(Pro)42 b(of.)663 b Fj(F)-108 b(or)431 b Fh(k)414 b Fj(=)369 b(9,)433 b(w)-36 b(e)431 b(need)g(to)h(distribute)e Fh(n)292 b Fg(\241)f Fh(j)444 b Fj(=)368 b(8.)578 b(The)432 b(partitions)f(of)i(8)e(that)g (do)h(not)f(con)-36 b(tain)800 5450 y(a)473 b(1)h(are)f Fg(f)p Fj(8)p Fg(g)p Fj(,)484 b Fg(f)p Fj(6)p Fh(;)221 b Fj(2)p Fg(g)p Fj(,)484 b Fg(f)p Fj(5)p Fh(;)221 b Fj(3)p Fg(g)p Fj(,)485 b Fg(f)p Fj(4)p Fh(;)221 b Fj(4)p Fg(g)p Fj(,)485 b Fg(f)p Fj(4)p Fh(;)221 b Fj(2)p Fh(;)g Fj(2)p Fg(g)p Fj(,)486 b Fg(f)p Fj(3)p Fh(;)221 b Fj(3)p Fh(;)g Fj(2)p Fg(g)475 b Fj(and)e Fg(f)p Fj(2)p Fh(;)221 b Fj(2)p Fh(;)g Fj(2)p Fh(;)g Fj(2)p Fg(g)476 b Fj(and)d(together)g(generate)g (a)800 7055 y(total)434 b(of)6907 9875 y Fh(j)370 b Fj(+)295 b(2)p Fh(j)75 b Fj(\()p Fh(j)369 b Fg(\241)296 b Fj(1\))f(+)15885 8002 y Fd(\265)16882 8976 y Fh(j)16863 10786 y Fj(2)17513 8002 y Fd(\266)18786 9875 y Fj(+)g(2)p Fh(j)21356 8002 y Fd(\265)22334 8976 y Fh(j)370 b Fg(\241)295 b Fj(1)23452 10786 y(2)25221 8002 y Fd(\266)26494 9875 y Fj(+)27800 8002 y Fd(\265)28797 8976 y Fh(j)28778 10786 y Fj(4)29429 8002 y Fd(\266)30775 9875 y Fj(=)369 b Fh(j)h Fj(+)295 b(5)35021 8002 y Fd(\265)36018 8976 y Fh(j)35999 10786 y Fj(2)36649 8002 y Fd(\266)37922 9875 y Fj(+)g(6)39879 8002 y Fd(\265)40876 8976 y Fh(j)40857 10786 y Fj(3)41508 8002 y Fd(\266)42781 9875 y Fj(+)44087 8002 y Fd(\265)45084 8976 y Fh(j)45065 10786 y Fj(4)45716 8002 y Fd(\266)800 13237 y Fj(comp)36 b(ositions.)584 b Fh(C)10024 13436 y Fl(j)10511 13237 y Fj(\()p Fh(j)371 b Fj(+)296 b(8)p Fh(;)14466 12953 y Fj(^)14466 13237 y(2)q(\))435 b(app)36 b(ears)435 b(as)h(A008779)h(in)f(Sloane's)f(On-Line)f(Encyclop)36 b(edia)436 b(of)g(In)-36 b(teger)800 14842 y(Sequences)433 b([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.10) [[137 583 149 595] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)579 b(A008779)435 b(is)f(de\257ned)e(b)-36 b(y)15374 17994 y Fh(a)p Fj(\()p Fh(n)p Fj(\))369 b(=)f(1)296 b(+)e(6)p Fh(n)i Fj(+)f(11)26175 16121 y Fd(\265)27154 17096 y Fh(n)27217 18905 y Fj(2)27930 16121 y Fd(\266)29203 17994 y Fj(+)g(7)31160 16121 y Fd(\265)32138 17096 y Fh(n)32201 18905 y Fj(3)32914 16121 y Fd(\266)34187 17994 y Fj(+)35494 16121 y Fd(\265)36472 17096 y Fh(n)36535 18905 y Fj(4)37249 16121 y Fd(\266)800 21146 y Fj(and)576 b(coun)-36 b(ts)575 b(the)h Fh(n)p Fj(-dimensional)g(partitions)g(of)h (5)g(\(for)f(a)h(de\257nition)e(see)i([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.1) [[431 527 437 539] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)613 b(p.)1005 b(179\).)j(W)-108 b(e)576 b(need)800 22751 y(to)566 b(sho)-36 b(w)566 b(the)g(equiv)-72 b(alence)566 b(of)h(the)e(sequences)h Fh(a)p Fj(\()p Fh(n)p Fj(\))g(and)f Fh(C)31897 22950 y Fl(j)32384 22751 y Fj(\()p Fh(j)460 b Fj(+)385 b(8)p Fh(;)36517 22467 y Fj(^)36517 22751 y(2)q(\))566 b(for)g(a)g(suitable)g(v)-72 b(alue)567 b(of)f Fh(n)p Fj(.)800 24356 y(Replacing)443 b Fh(j)517 b Fj(b)-36 b(y)443 b Fh(j)376 b Fj(+)301 b(1)442 b(in)h(the)e (expression)i(deriv)-36 b(ed)442 b(ab)36 b(o)-36 b(v)g(e)443 b(for)g Fh(C)34245 24555 y Fl(j)34732 24356 y Fj(\()p Fh(j)376 b Fj(+)300 b(8)p Fh(;)38696 24072 y Fj(^)38696 24356 y(2)r(\))442 b(and)g(simplifying)i(sho)-36 b(ws)800 25961 y(that)433 b Fh(C)4549 26160 y Fl(j)51 b Fc(+1)6238 25961 y Fj(\(\()p Fh(j)370 b Fj(+)294 b(1\))i(+)f(8)p Fh(;)13455 25677 y Fj(^)13455 25961 y(2)q(\))369 b(=)f Fh(a)p Fj(\()p Fh(j)75 b Fj(\),)433 b(i.e.,)i Fh(C)22851 26160 y Fl(j)23338 25961 y Fj(\()p Fh(j)369 b Fj(+)295 b(8)p Fh(;)27290 25677 y Fj(^)27290 25961 y(2)r(\))368 b(=)h Fh(a)p Fj(\()p Fh(j)h Fg(\241)295 b Fj(1\),)434 b(similar)g(to)g(the)f(case)h Fh(k)414 b Fj(=)369 b(7.)p 52170 25961 630 630 v 800 29172 a Ff(Remark.)845 b Fj(The)486 b Fh(n)p Fj(-dimensional)g(partitions)g(of)h(4)f(and)f(5)i(are)f(not)f (the)h(only)g(ones)g(con)-36 b(tained)486 b(in)g(the)800 30777 y(diagonals)427 b(of)f(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 426 w(3)p (#table.3) [[176 440 182 452] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)576 b(F)-108 b(urther)424 b(study)h(sho)-36 b(ws)426 b(that)g(the)f Fh(n)p Fj(-dimensional)h(partitions)f(of)i(2)f(and)f(3)h (also)800 32382 y(o)36 b(ccur)484 b(on)g(the)f(diagonals.)731 b(In)483 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.1) [[224 426 230 438] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)497 b(form)-36 b(ulas)484 b(for)h(the)e Fh(n)p Fj(-dimensional)h(partitions)g(of)h Fh(k)529 b Fj(are)484 b(giv)-36 b(en)484 b(for)800 33987 y Fh(k)f Fg(\267)438 b Fj(6.)699 b(The)474 b Fh(n)p Fj(-dimensional)g (partitions)g(of)h(2)f(are)g(giv)-36 b(en)475 b(as)f Fh(a)p Fj(\()p Fh(n)p Fj(\))438 b(=)f Fh(n)323 b Fj(+)f(1,)485 b(whic)-36 b(h)473 b(is)i(the)e(sequence)800 35592 y(in)454 b(the)g(3)5249 35110 y Fc(rd)6648 35592 y Fj(diagonal:)621 b Fh(C)13329 35791 y Fl(j)13816 35592 y Fj(\()p Fh(j)384 b Fj(+)309 b(2)p Fh(;)17797 35308 y Fj(^)17797 35592 y(2)q(\))404 b(=)g Fh(a)p Fj(\()p Fh(j)384 b Fg(\241)309 b Fj(1\).)641 b(The)455 b Fh(n)p Fj(-dimensional)f(partitions)g(of)i(3) e(are)h(giv)-36 b(en)455 b(as)800 37197 y Fh(a)p Fj(\()p Fh(n)p Fj(\))477 b(=)f(1)339 b(+)f(2)p Fh(n)h Fj(+)f Fh(n)p Fj(\()p Fh(n)h Fg(\241)f Fj(1\))p Fh(=)p Fj(2)498 b(,)513 b(and)497 b(simple)g(algebraic)h(manipulation)f(sho)-36 b(ws)497 b(that)f(this)h(sequence)800 38802 y(app)36 b(ears)535 b(on)g(the)g(5)10614 38320 y Fc(th)12093 38802 y Fj(diagonal:)783 b Fh(C)18936 39001 y Fl(j)19422 38802 y Fj(\()p Fh(j)439 b Fj(+)364 b(4)p Fh(;)23513 38518 y Fj(^)23513 38802 y(2)r(\))541 b(=)h Fh(a)p Fj(\()p Fh(j)438 b Fg(\241)365 b Fj(1\).)883 b(A)535 b(pattern)f(emerges:)782 b(for)536 b(o)36 b(dd)535 b Fh(k)45 b Fj(,)561 b(the)800 40407 y Fh(k)1521 39925 y Fc(th)2983 40407 y Fj(diagonal)518 b(con)-36 b(tains)518 b(the)f Fh(n)p Fj(-dimensional)h(partitions)f(of) h(\()p Fh(k)398 b Fj(+)352 b(1\))p Fh(=)p Fj(2.)831 b(This)518 b(conjecture)f(w)-36 b(as)518 b(v)-36 b(ery)800 42012 y(exciting)458 b(b)36 b(ecause)457 b(no)g(generating)h(function)e (exists)i(for)g(this)f(family;)471 b(if)458 b(a)f(nice)g(connection)g (could)g(b)36 b(e)800 43617 y(established,)616 b(then)579 b(one)h(could)g(compute)f(the)g Fh(n)p Fj(-dimensional)h(partitions)g (in)f(a)h(simple)g(w)-36 b(a)g(y)-108 b(,)617 b(using)800 45222 y(Theorem)p 0 1 0 0 TeXcolorcmyk 546 w(4)p (#theorem.4) [[130 310 135 322] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)914 b(Ho)-36 b(w)g(ev)g(er,)574 b(the)545 b(pattern)f(do)36 b(es)546 b(not)f(con)-36 b(tin)g(ue:)801 b(the)545 b(sequence)g(for)h (the)f Fh(n)p Fj(-dimensional)800 46828 y(partitions)471 b(of)g(6,)481 b(namely)-108 b(,)481 b Fg(f)p Fj(11)p Fh(;)221 b Fj(48)p Fh(;)g Fj(140)p Fh(;)g Fj(326)p Fh(;)g Fj(657)p Fh(;)g Fj(1197)p Fh(;)g(:)g(:)g(:)11 b Fg(g)p Fj(,)481 b(whic)-36 b(h)470 b(should)h(ha)-36 b(v)g(e)471 b(app)36 b(eared)470 b(in)h(the)800 48433 y(11)2100 47951 y Fc(th)3529 48433 y Fj(d)484 b(iagonal,)499 b(do)36 b(es)484 b(not)h(sho)-36 b(w)484 b(up.)730 b(Nor)485 b(do)36 b(es)484 b(this)g(sequence)h(app)36 b(ear)484 b(in)g(the)g(13)43937 47951 y Fc(th)45366 48433 y Fj(diagonal,)499 b(the)800 50038 y(only)434 b(one)g(that)f(has)g(11)i(as)f(the)f(second) g(elemen)-36 b(t.)800 54394 y Fk(5)2152 b(The)716 b(n)-60 b(um)g(b)60 b(er)716 b(of)h(palindromes)f(with)g(no)h(2's)800 57315 y Fj(As)476 b(a)g(last)h(exploration,)488 b(let)476 b(us)f(consider)h(the)f(n)-36 b(um)g(b)36 b(er)475 b(of)i(palindromes)e (of)i Fh(n)f Fj(with)g(no)g(2's.)706 b(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 477 w(4)p (#table.4) [[541 201 547 213] [1 1 1 [3 3]] [0 0 1]] pdfm Black 800 58920 a(lists)434 b(the)f(actual)h(palindromes)g(for)g(the)f (\257rst)f(few)j(v)-72 b(alues)434 b(of)g Fh(n)p Fj(.)2751 60525 y(In)613 b(the)g(follo)-36 b(wing)615 b(theorem,)658 b(w)-36 b(e)613 b(giv)-36 b(e)614 b(recursiv)-36 b(e)613 b(form)-36 b(ulas)614 b(and)f(the)f(generating)i(function)e(for)800 62130 y Fh(P)181 b Fj(\()p Fh(n;)3683 61846 y Fj(^)3683 62130 y(2)q(\).)p Black 800 64375 a Ff(Theorem)499 b(10)p Black 651 w Fi(The)465 b(numb)-66 b(er)463 b(of)i(p)-66 b(alindr)g(omes)464 b(of)g Fh(n)i Fi(without)f(2's)g(is)g(given)e(by) 8609 67547 y Fh(P)181 b Fj(\()p Fh(n;)11492 67263 y Fj(^)11492 67547 y(2)q(\))369 b(=)14399 65674 y Fd(\275)15948 66733 y Fh(C)95 b Fj(\()p Fh(k)340 b Fj(+)295 b(1)p Fh(;)21035 66449 y Fj(^)21035 66733 y(2)r(\))p Fh(;)9439 b Fi(for)465 b Fh(n)369 b Fj(=)g(2)p Fh(k)45 b(;)221 b(k)415 b Fg(\270)369 b Fj(0;)15948 68421 y Fh(C)95 b Fj(\()p Fh(k)340 b Fj(+)295 b(1)p Fh(;)21035 68137 y Fj(^)21035 68421 y(2)r(\))g(+)f Fh(C)95 b Fj(\()p Fh(k)340 b Fg(\241)296 b Fj(1)p Fh(;)28903 68137 y Fj(^)28903 68421 y(2)q(\))p Fh(;)1572 b Fi(for)465 b Fh(n)369 b Fj(=)g(2)p Fh(k)340 b Fj(+)295 b(1)p Fh(;)221 b(k)415 b Fg(\270)369 b Fj(0)p Fh(;)800 71101 y Fi(with)465 b(gener)-66 b(ating)462 b(function)h Fh(G)16056 71300 y Fl(P)16839 71101 y Fj(\()p Fh(z)59 b Fj(\))369 b(=)20263 70104 y Fd(P)21665 70455 y Fe(1)21665 71488 y Fl(n)p Fc(=0)23715 71101 y Fh(P)181 b Fj(\()p Fh(n;)26598 70817 y Fj(^)26598 71101 y(2)r(\))p Fh(z)28419 70619 y Fl(n)29414 71101 y Fj(=)32503 70578 y Fn(1)269 b(+)g Fm(z)p 30928 70795 5855 54 v 30928 71859 a Fn(1)g Fa(\241)g Fm(z)33632 71419 y Fc(2)34427 71859 y Fa(\241)g Fm(z)36256 71419 y Fc(3)36915 71101 y Fi(.)p Black 26475 74617 a Fj(9)p Black eop %%Page: 10 10 10 9 bop Black 0 TeXcolorgray Black Black Black Black Black 7093 1579 39415 45 v 7071 3184 45 1606 v 10757 2703 a Fh(n)p 15176 3184 V 4218 w Fj(1)p 16933 3184 V 2234 w(2)p 20942 3184 V 4485 w(3)p 27204 3184 V 6739 w(4)p 35719 3184 V 8990 w(5)p 46485 3184 V 7093 3228 39415 45 v 7071 4834 45 1606 v 7646 4352 a(P)-36 b(alindromes)p 15176 4834 V 1107 w(1)p 16933 4834 V 1107 w(1)296 b(+)f(1)p 20942 4834 V 1107 w(1)h(+)e(1)i(+)f(1)p 27204 4834 V 1107 w(1)h(+)e(1)i(+)f(1)g(+)g(1)p 35719 4834 V 1107 w(1)h(+)f(1)g(+)g(1)g(+)g(1)h(+)f(1)p 46485 4834 V 7071 6439 V 10016 5957 a(of)435 b Fh(n)p 15176 6439 V 16933 6439 V 20942 6439 V 11496 w Fj(3)p 27204 6439 V 6739 w(4)p 35719 6439 V 6738 w(1)295 b(+)g(3)g(+)g(1)p 46485 6439 V 7071 8044 V 7998 7562 a(with)434 b(no)f(2's)p 15176 8044 V 16933 8044 V 20942 8044 V 27204 8044 V 35719 8044 V 26508 w(5)p 46485 8044 V 7093 8088 39415 45 v Black 16008 11667 a(T)-108 b(able)434 b(4:)p 0 TeXcolorgray Black 579 w(P)-36 b(alindromes)434 b(of)g Fh(n)g Fj(with)f(no)h(2's)p Black Black 800 15134 a Ff(Pro)42 b(of.)948 b Fj(In)515 b(general,)538 b(for)516 b(o)36 b(dd)516 b Fh(n)p Fj(,)537 b(b)36 b(egin)516 b(with)g(an)-36 b(y)516 b(o)36 b(dd)515 b(n)-36 b(um)g(b)36 b(er)514 b(1)c Fg(\267)f Fh(m)g Fg(\267)g Fh(n)516 b Fj(as)h(a)f(middle)f(en)-36 b(try)800 16739 y(and)603 b(\257ll)h(in)g(the)f(left)h(side)g(of)g(the)f(palindrome)h (of)g Fh(n)g Fj(with)g(an)-36 b(y)604 b(comp)36 b(osition)604 b(of)h(\()p Fh(n)411 b Fg(\241)g Fh(m)p Fj(\))p Fh(=)p Fj(2)659 b(=)f Fh(j)800 18344 y Fj(that)548 b(has)h(no)f(2's)i(and)e (complete)h(the)f(righ)-36 b(t)548 b(side)h(of)g(the)g(palindrome)f(of) h Fh(n)g Fj(with)g(the)f(comp)36 b(osition)800 19949 y(of)532 b Fh(j)607 b Fj(in)531 b(opp)36 b(osite)532 b(order.)871 b(The)532 b(total)g(n)-36 b(um)g(b)36 b(er)530 b(of)i(suc)-36 b(h)531 b(palindromes)g(is)h(giv)-36 b(en)532 b(b)-36 b(y)531 b Fh(P)181 b Fj(\(2)p Fh(k)407 b Fj(+)362 b(1)p Fh(;)50096 19665 y Fj(^)50096 19949 y(2)q(\))535 b(=)800 20558 y Fd(P)2202 20908 y Fl(k)2202 21942 y(j)51 b Fc(=0)4113 21554 y Fh(C)95 b Fj(\()p Fh(j)r(;)6767 21270 y Fj(^)6767 21554 y(2)q(\))470 b(=)f Fh(C)95 b Fj(\()p Fh(k)380 b Fj(+)335 b(1)p Fh(;)15042 21270 y Fj(^)15042 21554 y(2)r(\))g(+)g Fh(C)95 b Fj(\()p Fh(k)381 b Fg(\241)335 b Fj(1)p Fh(;)23071 21270 y Fj(^)23071 21554 y(2)r(\).)755 b(F)-108 b(or)493 b(ev)-36 b(en)492 b Fh(n)p Fj(,)509 b(w)-36 b(e)493 b(m)-36 b(ust)492 b(omit)h(the)f (palindromes)g(that)800 23159 y(are)482 b(formed)h(with)f(a)g(2)g(in)g (the)g(middle,)494 b(but)481 b(do)h(allo)-36 b(w)484 b(an)e(ev)-36 b(en)482 b(split,)495 b(i.e.,)g(no)482 b(middle)g(term.)723 b(Th)-36 b(us,)800 24764 y Fh(P)181 b Fj(\(2)p Fh(k)45 b(;)4278 24480 y Fj(^)4278 24764 y(2)q(\))369 b(=)g Fh(C)95 b Fj(\()p Fh(k)45 b(;)10020 24480 y Fj(^)10020 24764 y(2)q(\))153 b(+)12495 23768 y Fd(P)13895 24119 y Fl(k)24 b Fe(\241)p Fc(2)13895 25152 y Fl(j)51 b Fc(=0)15888 24764 y Fh(C)95 b Fj(\()p Fh(j)r(;)18542 24480 y Fj(^)18542 24764 y(2)q(\))369 b(=)g Fh(C)95 b Fj(\()p Fh(k)197 b Fj(+)153 b(1)p Fh(;)26251 24480 y Fj(^)26251 24764 y(2)q(\),)377 b(where)364 b(the)f(last)i(equalit)-36 b(y)364 b(follo)-36 b(ws)366 b(from)e(Eq.)h(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[534 494 539 506] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)800 26641 y(Note)434 b(that)f(w)-36 b(e)434 b(de\257ne)e Fh(P)181 b Fj(\(0)p Fh(;)15190 26357 y Fj(^)15190 26641 y(2)q(\))369 b(=)g(1)434 b(similar)g(to)g(the)f(de\257nition)g (for)h Fh(C)95 b Fj(\(0)p Fh(;)37813 26357 y Fj(^)37813 26641 y(2)q(\).)2751 28246 y(T)-108 b(o)351 b(deriv)-36 b(e)351 b(the)f(generating)g(function,)368 b(w)-36 b(e)350 b(separate)h Fh(G)30064 28445 y Fl(P)30847 28246 y Fj(\()p Fh(z)59 b Fj(\))350 b(in)-36 b(to)351 b(o)36 b(dd)350 b(and)g(ev)-36 b(en)350 b(terms,)367 b(substitute)800 29851 y(the)437 b(relev)-72 b(an)-36 b(t)437 b(form)-36 b(ulas,)440 b(factor)e(out)f(appropriate)g(p)36 b(o)-36 b(w)g(ers)437 b(of)h Fh(z)497 b Fj(and)437 b(express)g(the)g(resulting) g(series)h(in)800 31457 y(terms)433 b(of)i(the)e(generating)g(function) h Fh(G)20621 31656 y Fl(C)21409 31457 y Fj(\()p Fh(z)22578 30974 y Fc(2)23104 31457 y Fj(\):)8196 35270 y Fh(G)9222 35469 y Fl(P)10005 35270 y Fj(\()p Fh(z)59 b Fj(\))1107 b(=)15394 33609 y Fe(1)14905 34008 y Fd(X)15007 36837 y Fl(k)24 b Fc(=0)17045 35270 y Fh(P)181 b Fj(\(2)p Fh(k)341 b Fj(+)295 b(1)p Fh(;)22776 34986 y Fj(^)22776 35270 y(2)q(\))p Fh(z)24596 34721 y Fc(2)p Fl(k)24 b Fc(+1)27133 35270 y Fj(+)28929 33609 y Fe(1)28440 34008 y Fd(X)28541 36837 y Fl(k)g Fc(=0)30580 35270 y Fh(P)181 b Fj(\(2)p Fh(k)45 b(;)34058 34986 y Fj(^)34058 35270 y(2)q(\))p Fh(z)35878 34721 y Fc(2)p Fl(k)12787 39587 y Fj(=)15045 38688 y(1)p 15038 39281 664 54 v 15038 40498 a Fh(z)16545 37927 y Fe(1)16056 38325 y Fd(X)16157 41154 y Fl(k)24 b Fc(=0)18196 39587 y Fh(C)95 b Fj(\()p Fh(k)340 b Fj(+)295 b(1)p Fh(;)23283 39303 y Fj(^)23283 39587 y(2)q(\)\()p Fh(z)25609 39039 y Fc(2)26135 39587 y Fj(\))26641 39039 y Fl(k)24 b Fc(+1)28707 39587 y Fj(+)295 b Fh(z)30677 39039 y Fc(3)31913 37927 y Fe(1)31424 38325 y Fd(X)31526 41154 y Fl(k)24 b Fc(=0)33564 39587 y Fh(C)95 b Fj(\()p Fh(k)340 b Fg(\241)296 b Fj(1)p Fh(;)38673 39303 y Fj(^)38673 39587 y(2)q(\)\()p Fh(z)40999 39039 y Fc(2)41525 39587 y Fj(\))42031 39039 y Fl(k)24 b Fe(\241)p Fc(1)44097 39587 y Fj(+)15307 43006 y(1)p 15038 43599 1190 54 v 15038 44815 a Fh(z)15701 44432 y Fc(2)17070 42244 y Fe(1)16581 42642 y Fd(X)16683 45471 y Fl(k)g Fc(=0)18722 43904 y Fh(C)95 b Fj(\()p Fh(k)340 b Fj(+)294 b(1)p Fh(;)23808 43620 y Fj(^)23808 43904 y(2)r(\)\()p Fh(z)26135 43356 y Fc(2)26661 43904 y Fj(\))27167 43356 y Fl(k)24 b Fc(+1)12787 47998 y Fj(=)1106 b Fh(G)15931 48197 y Fl(C)16720 47998 y Fj(\()p Fh(z)17889 47450 y Fc(2)18415 47998 y Fj(\))19142 46125 y Fd(\265)20253 47100 y Fh(z)354 b Fj(+)295 b Fh(z)23181 46617 y Fc(5)24002 47100 y Fj(+)g(1)p 20253 47693 5707 54 v 22511 48909 a Fh(z)23174 48526 y Fc(2)26092 46125 y Fd(\266)27365 47998 y Fg(\241)28833 47100 y Fj(1)p 28827 47693 664 54 v 28827 48909 a Fh(z)29918 47998 y Fg(\241)31649 47100 y Fj(1)p 31379 47693 1190 54 v 31379 48909 a Fh(z)32042 48526 y Fc(2)32701 47998 y Fh(:)800 51620 y Fj(Substituting)432 b(the)h(form)-36 b(ula)434 b(for)g Fh(G)18272 51819 y Fl(C)19061 51620 y Fj(\()p Fh(z)20230 51137 y Fc(2)20756 51620 y Fj(\))f(and)g(simplifying)i(giv) -36 b(es)435 b(the)e(desired)g(result.)p 52170 51620 630 630 v 2751 54830 a(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 434 w(5)p (#table.5) [[129 224 135 236] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(giv)-36 b(es)434 b(the)f(n)-36 b(um)g(b)36 b(er)432 b(of)j(palindromes)e(of)h Fh(n)g Fj(without)g(2's)g(for)g(0) 369 b Fg(\267)h Fh(n)f Fg(\267)g Fj(16.)p Black Black Black 7013 58045 39574 45 v 6991 59650 45 1606 v 9199 59169 a Fh(n)p 12138 59650 V 2739 w Fj(0)p 13896 59650 V 1107 w(1)p 15653 59650 V 1107 w(2)p 17410 59650 V 1108 w(3)p 19167 59650 V 1107 w(4)p 20925 59650 V 1107 w(5)p 22682 59650 V 1108 w(6)p 24439 59650 V 1107 w(7)p 26197 59650 V 1107 w(8)p 27954 59650 V 1108 w(9)p 29711 59650 V 1107 w(10)p 32119 59650 V 1107 w(11)p 34526 59650 V 1108 w(12)p 36934 59650 V 1108 w(13)p 39342 59650 V 1107 w(14)p 41749 59650 V 1108 w(15)p 44157 59650 V 1108 w(16)p 46565 59650 V 7013 59695 39574 45 v 6991 61383 45 1689 v 7567 60901 a Fh(P)181 b Fj(\()p Fh(n;)10450 60617 y Fj(^)10450 60901 y(2)q(\))p 12138 61383 V 1107 w(1)p 13896 61383 V 1107 w(1)p 15653 61383 V 1107 w(1)p 17410 61383 V 1108 w(2)p 19167 61383 V 1107 w(2)p 20925 61383 V 1107 w(3)p 22682 61383 V 1108 w(4)p 24439 61383 V 1107 w(5)p 26197 61383 V 1107 w(7)p 27954 61383 V 1108 w(9)p 29711 61383 V 1107 w(12)p 32119 61383 V 1107 w(16)p 34526 61383 V 1108 w(21)p 36934 61383 V 1108 w(28)p 39342 61383 V 1107 w(37)p 41749 61383 V 1108 w(49)p 44157 61383 V 1108 w(65)p 46565 61383 V 7013 61427 39574 45 v Black 11591 65006 a(T)-108 b(able)434 b(5:)p 0 TeXcolorgray Black 579 w(The)g(n)-36 b(um)g(b)36 b(er)432 b(of)i(palindromes)f(of)i Fh(n)f Fj(without)f(2's)p Black Black 2751 68086 a(The)630 b(sequence)h Fh(P)181 b Fj(\(2)p Fh(k)474 b Fj(+)429 b(1)p Fh(;)17207 67801 y Fj(^)17207 68086 y(2)q(\))631 b(app)36 b(ears)630 b(as)h(A005314)h(in)e([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.10) [[372 104 384 116] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])h(as)g(one)f(of)h(the)f(sequences)h(used)800 69691 y(to)j(de\257ne)f(T)-108 b(o)36 b(eplitz)634 b(matrices)g(whose)h (in)-36 b(v)g(erses)634 b(con)-36 b(tain)633 b(large)i(en)-36 b(tries)633 b(\(for)i(a)f(de\257nition)f(see)h([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.8) [[535 90 541 102] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)800 71296 y(p.)1158 b(130\).)h(A005314's)629 b(de\257nition,)675 b Fh(a)p Fj(\()p Fh(n)p Fj(\))697 b(=)h(2)p Fh(a)p Fj(\()p Fh(n)427 b Fg(\241)g Fj(1\))g Fg(\241)g Fh(a)p Fj(\()p Fh(n)g Fg(\241)g Fj(2\))g(+)f Fh(a)p Fj(\()p Fh(n)h Fg(\241)g Fj(3\))627 b(is)g(iden)-36 b(tical)627 b(to)p Black 26150 74617 a(10)p Black eop %%Page: 11 11 11 10 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Equation)p 0 1 0 0 TeXcolorcmyk 636 w(2)p (#equation.2) [[132 704 137 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)687 b(whic)-36 b(h)636 b(coun)-36 b(ts)635 b(the)g(n)-36 b(um)g(b)36 b(er)634 b(of)j(comp)36 b(ositions.)1186 b(There)636 b(is)g(an)f(easy)i(com)-36 b(binatorial)800 3029 y(explanation)413 b(for)f(this)g(fact,)417 b(namely)c(an)f (alternativ)-36 b(e)413 b(metho)36 b(d)411 b(to)i(create)f (palindromes.)571 b(Rather)412 b(than)800 4634 y(using)575 b(comp)36 b(ositions,)612 b(w)-36 b(e)576 b(pro)36 b(ceed)576 b(in)f(a)h(manner)f(similar)h(to)g(the)f(w)-36 b(a)g(y)576 b(w)-36 b(e)576 b(created)f(comp)36 b(ositions)800 6239 y(recursiv)-36 b(ely:)697 b(App)36 b(end)490 b(\\1+")j(and)f(\\+1")h (to)f(the)g(left)g(and)g(righ)-36 b(t)492 b(end)f(of)i(a)g(palindrome)f (of)h(\(o)36 b(dd\))491 b Fh(n)p Fj(,)800 7844 y(resp)36 b(ectiv)-36 b(ely)-108 b(,)455 b(or)449 b(increase)h(the)f(t)-36 b(w)g(o)450 b(end)f(summands)g(b)-36 b(y)449 b(1.)627 b(If)451 b(the)e(palindrome)g(consists)h(of)g(a)g(single)800 9450 y(summand,)422 b(increase)f(the)f(summand)f(b)-36 b(y)420 b(2)h(instead.)574 b(Delete)421 b(the)f(palindromes)g(that)g(w) -36 b(ould)420 b(result)g(in)800 11055 y(a)439 b(forbidden)e(2,)j(and)e (create)g(those)g(that)g(ha)-36 b(v)g(e)438 b(end)g(summands)f(3)h(b) -36 b(y)438 b(adding)g(a)h(3)f(to)h(b)36 b(oth)437 b(ends)h(of)h(a)800 12660 y(palindrome)468 b(of)g Fh(n)319 b Fg(\241)g Fj(6.)681 b(Th)-36 b(us,)476 b(the)467 b(total)h(n)-36 b(um)g(b)36 b(er)466 b(of)j(palindromes)e(for)i(o)36 b(dd)467 b Fh(n)h Fj(is)g(giv)-36 b(en)469 b(b)-36 b(y)467 b(is)h(giv)-36 b(en)800 14265 y(b)g(y)418 b Fh(P)181 b Fj(\(2)p Fh(k)310 b Fj(+)264 b(1)p Fh(;)8260 13981 y Fj(^)8260 14265 y(2)q(\))369 b(=)f(2)p Fh(P)181 b Fj(\(2\()p Fh(k)310 b Fg(\241)265 b Fj(1\))f(+)g(1)p Fh(;)20709 13981 y Fj(^)20709 14265 y(2)q(\))g Fg(\241)h Fh(P)181 b Fj(\(2\()p Fh(k)309 b Fg(\241)264 b Fj(2\))h(+)f(1)p Fh(;)32320 13981 y Fj(^)32320 14265 y(2)q(\))g(+)g Fh(P)181 b Fj(\(2\()p Fh(k)309 b Fg(\241)265 b Fj(3\))f(+)g(1)p Fh(;)43909 13981 y Fj(^)43909 14265 y(2)q(\).)573 b(Comparison)800 15870 y(of)434 b(the)f(initial)i (terms)e(\()p Fh(a)p Fj(\(0\))369 b(=)f(0)p Fh(;)656 b(a)p Fj(\(1\))368 b(=)h(1)p Fh(;)656 b(a)p Fj(\(2\))368 b(=)h(2\))434 b(sho)-36 b(ws)433 b(that)h Fh(P)181 b Fj(\(2)p Fh(k)340 b Fj(+)295 b(1)p Fh(;)41895 15586 y Fj(^)41895 15870 y(2)q(\))369 b(=)g Fh(a)p Fj(\()p Fh(k)339 b Fj(+)295 b(1\).)2751 17475 y(The)372 b(complete)h(sequence)f Fh(P)181 b Fj(\()p Fh(n;)19006 17191 y Fj(^)19006 17475 y(2)q(\))372 b(app)36 b(ears)372 b(in)g([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.10) [[316 560 328 572] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])h(as)f(shifted)g(sequence)g(A000931,)387 b(the)372 b(P)-36 b(ado)g(v)-72 b(an)800 19080 y(sequence,)567 b(with)539 b Fh(a)p Fj(\()p Fh(n)p Fj(\))550 b(=)f Fh(P)181 b Fj(\()p Fh(n)368 b Fg(\241)g Fj(5)p Fh(;)19637 18796 y Fj(^)19637 19080 y(2)q(\),)566 b(and)540 b(as)g(A078027,)568 b(with)540 b Fh(a)p Fj(\()p Fh(n)p Fj(\))550 b(=)f Fh(P)181 b Fj(\()p Fh(n)368 b Fg(\241)f Fj(7)p Fh(;)44816 18796 y Fj(^)44816 19080 y(2)r(\).)897 b(Using)540 b(the)800 20685 y(generating)352 b(function)g(deriv)-36 b(ed)351 b(in)h(Theorem)p 0 1 0 0 TeXcolorcmyk 352 w(10)p (#theorem.10) [[283 531 294 543] [1 1 1 [3 3]] [0 0 1]] pdfm Black 353 w(and)f(standard)g(metho)36 b(ds)351 b(to)h(compute)f(the)h (generating)800 22290 y(function)527 b(of)i(a)e(shifted)h(sequence)f (\(see)h(for)g(example)g([)p 0 1 0 0 TeXcolorcmyk(11)p (#cite.W) [[332 516 344 528] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)552 b(Rule)527 b(1,)552 b(p.)860 b(34\),)552 b(it)527 b(can)h(b)36 b(e)527 b(established)800 23895 y(that)433 b(the)g(sequences)h(are)g(iden)-36 b(tical.)800 28332 y Fk(6)2152 b(Extensions)716 b(and)g(op)60 b(en)716 b(problems)800 31253 y Fj(W)-108 b(e)558 b(ha)-36 b(v)g(e)558 b(not)g(coun)-36 b(ted)557 b(the)g(n)-36 b(um)g(b)36 b(er)557 b(of)i(o)36 b(ccurrences)557 b(of)i(the)e(in)-36 b(teger)558 b Fh(i)g Fj(in)g(all)h(palindromes)f(of)g Fh(n)p Fj(,)800 32858 y(nor)595 b(the)f(n)-36 b(um)g(b)36 b(er)594 b(of)i(palindromes)f(of)g Fh(n)h Fj(with)f Fh(j)670 b Fj(parts.)1062 b(Previous)596 b(exp)36 b(erience)595 b([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.6) [[466 421 471 433] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(predicts)f(that)800 34463 y(the)455 b(form)-36 b(ulas)456 b(tend)f(to)g(b)36 b(e)456 b(more)f(complicated)h(for)g (palindromes,)462 b(as)455 b(it)h(is)g(necessary)g(to)g(distinguish)800 36068 y(b)36 b(et)-36 b(w)g(een)414 b(o)36 b(dd)413 b(and)h(ev)-36 b(en)414 b Fh(n)p Fj(,)419 b(as)c(seen)e(also)j(in)e(Theorem)g(10)h (for)f(the)g(total)h(n)-36 b(um)g(b)36 b(er)412 b(of)j(palindromes)f (of)800 37673 y Fh(n)p Fj(.)2751 39278 y(Another)398 b(problem)g(is)i(to)e(ask)i(similar)g(questions)f(for)g(partitions)g (of)g Fh(n)h Fj(without)e(2's.)568 b(In)398 b(the)g(pro)36 b(ofs)800 40884 y(of)420 b(Theorems)g(5)f(through)f(9)i(w)-36 b(e)420 b(used)e(partitions)h(in)g(order)g(to)h(coun)-36 b(t)418 b(the)h(comp)36 b(ositions)420 b(without)f(2's.)800 42489 y(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 477 w(6)p (#table.6) [[112 335 118 347] [1 1 1 [3 3]] [0 0 1]] pdfm Black 477 w(giv)-36 b(es)478 b(the)e(n)-36 b(um)g(b)36 b(er)475 b(of)j(partitions)f(of)g Fh(n)g Fj(with)g(no)g(2's,)488 b(computed)476 b(using)g Fi(Mathematic)-66 b(a)p Fj(.)707 b(No)800 44094 y(easy)316 b(recursion)f(exists)g(to)g(create)h(the)e (partitions)h(of)h Fh(n)53 b Fj(+)g(1)315 b(from)h(those)e(of)i Fh(n)p Fj(.)539 b(Again)316 b(the)e(corresp)36 b(onding)800 45699 y(sequence)434 b(is)f(not)h(y)-36 b(et)434 b(listed)f(in)h([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.10) [[228 306 240 318] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)p Black Black Black 10383 49064 32835 45 v 10361 50669 45 1606 v 11835 50188 a Fh(n)p 14041 50669 V 2006 w Fj(1)p 15798 50669 V 1107 w(2)p 17556 50669 V 1107 w(3)p 19313 50669 V 1107 w(4)p 21070 50669 V 1108 w(5)p 22827 50669 V 1107 w(6)p 24585 50669 V 1107 w(7)p 26342 50669 V 1433 w(8)p 28750 50669 V 1757 w(9)p 31157 50669 V 1433 w(10)p 33565 50669 V 1107 w(11)p 35972 50669 V 1108 w(12)p 38380 50669 V 1108 w(13)p 40788 50669 V 1107 w(14)p 43195 50669 V 10383 50713 32835 45 v 10361 52318 45 1606 v 10936 51837 a Fh(\274)48 b Fj(\()p Fh(n)p Fj(\))p 14041 52318 V 1107 w(1)p 15798 52318 V 1107 w(1)p 17556 52318 V 1107 w(2)p 19313 52318 V 1107 w(3)p 21070 52318 V 1108 w(4)p 22827 52318 V 1107 w(6)p 24585 52318 V 1107 w(8)p 26342 52318 V 1108 w(11)p 28750 52318 V 1107 w(15)p 31157 52318 V 1108 w(20)p 33565 52318 V 1107 w(26)p 35972 52318 V 1108 w(35)p 38380 52318 V 1108 w(45)p 40788 52318 V 1107 w(58)p 43195 52318 V 10383 52363 32835 45 v 10320 53846 32960 45 v 10298 55451 45 1606 v 11772 54970 a Fh(n)p 13979 55451 V 2006 w Fj(15)p 16386 55451 V 1108 w(16)p 18794 55451 V 1433 w(17)p 21852 55451 V 1758 w(18)p 24910 55451 V 1758 w(19)p 27968 55451 V 1758 w(20)p 31026 55451 V 1758 w(21)p 34084 55451 V 1758 w(22)p 37142 55451 V 1757 w(23)p 40200 55451 V 1758 w(24)p 43258 55451 V 10320 55495 32960 45 v 10298 57100 45 1606 v 10874 56619 a Fh(\274)g Fj(\()p Fh(n)p Fj(\))p 13979 57100 V 1106 w(75)p 16386 57100 V 1108 w(96)p 18794 57100 V 1108 w(121)p 21852 57100 V 1108 w(154)p 24910 57100 V 1107 w(193)p 27968 57100 V 1108 w(242)p 31026 57100 V 1108 w(302)p 34084 57100 V 1108 w(375)p 37142 57100 V 1108 w(463)p 40200 57100 V 1108 w(573)p 43258 57100 V 10320 57145 32960 45 v Black 12278 60724 a(T)-108 b(able)434 b(6:)p 0 TeXcolorgray Black 578 w(The)g(n)-36 b(um)g(b)36 b(er)432 b(of)i(partitions)g(of)g Fh(n)g Fj(without)f(2's)p Black Black 800 66637 a Fk(7)2152 b(Ac)-60 b(kno)g(wledgemen)g(ts)800 69557 y Fj(The)608 b(authors)g(w)-36 b(ould)608 b(lik)-36 b(e)609 b(to)g(thank)f(the)f(anon)-36 b(ymous)608 b(referee)h(for)g (his)f(thorough)f(reading)i(of)g(the)800 71163 y(man)-36 b(uscript.)p Black 26150 74617 a(11)p Black eop %%Page: 12 12 12 11 bop Black 0 TeXcolorgray 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b(E.)h(Bergum)f(et)g (al.)h(\(eds.\),)3474 31695 y Fi(Applic)-66 b(ations)463 b(of)i(Fib)-66 b(onac)g(ci)462 b(Numb)-66 b(ers)p Fj(,)433 b(Klu)-36 b(w)g(er)433 b(Academic)h(Publishers,)g Ff(6)f Fj(\(1996\))i(165{171.)p Black 1450 34381 a([8])p Black 652 w(R.)419 b(L.)h(Graham)f(and)f(N.)i(J.)g(A.)f(Sloane,)k(An)-36 b(ti-Hadamard)418 b(matrices,)423 b Fi(Line)-66 b(ar)450 b(A)-33 b(lgebr)-66 b(a)451 b(Appl.)p Fj(,)422 b Ff(62)3474 35986 y Fj(\(1984\))434 b(113{137.)p Black 1450 38672 a([9])p Black 652 w(R.)627 b(P)-108 b(.)628 b(Grimaldi,)676 b(Comp)36 b(ositions)628 b(without)f(the)g(summand)g(1,)676 b Fi(Congr.)642 b(Numer.)626 b Ff(152)i Fj(\(2001\))3474 40277 y(33{43.)p Black 800 42963 a([10])p Black 652 w(N.)815 b(J.)g(A.)g(Sloane,)911 b(editor)814 b(\(2002\),)912 b Fi(The)815 b(On-Line)f(Encyclop)-66 b(e)g(dia)814 b(of)h(Inte)-66 b(ger)813 b(Se)-66 b(quenc)g(es)p Fj(,)3474 44568 y(h)-36 b(ttp://www.researc)g(h.att.com/)435 b(njas/sequences/)p Black 800 47253 a([11])p Black 652 w(H.)f(S.)f(Wilf,)i Fi(Gener)-66 b(atingfunctionolo)g(gy)p Fj(,)430 b(2)25165 46771 y Fl(nd)26709 47253 y Fj(edition,)k(Academic)g(Press,)g(1994.)p 800 50240 52000 45 v 800 52438 a(2000)h Fi(Mathematics)464 b(Subje)-66 b(ct)463 b(Classi\257c)-66 b(ation)p Fj(:)577 b(05A99)434 b(.)800 54043 y Fi(Keywor)-66 b(ds:)1963 b Fj(Comp)36 b(ositions,)667 b(palindromes,)f(n-dimensional)618 b(partitions,)666 b(p)36 b(en)-36 b(tagonal)620 b(p)-36 b(yramidal)800 55648 y(n)g(um)g(b)36 b(ers,)432 b(square)i(n)-36 b(um)g(b)36 b(ers,)433 b(triangle)h(n)-36 b(um)g(b)36 b(ers,)432 b(tilings.)p 800 58573 V 800 60845 a(\(Concerned)440 b(with)g(sequences)p 0 1 0 0 TeXcolorcmyk 440 w(A005251)p 16623 61058 4878 54 v [[222 169 266 181] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A005251) pdfm Black 2 w(,)p 0 1 0 0 TeXcolorcmyk 442 w(A008778)p 22303 61058 V [[273 169 317 181] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A008778) pdfm Black 2 w(,)p 0 1 0 0 TeXcolorcmyk 443 w(A002411)p 27984 61058 V [[324 169 368 181] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002411) pdfm Black 2 w(,)p 0 1 0 0 TeXcolorcmyk 442 w(A008779)p 33664 61058 V [[375 169 419 181] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A008779) pdfm Black 2 w(,)p 0 1 0 0 TeXcolorcmyk 442 w(A005314)p 39344 61058 V [[426 169 470 181] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A005314) pdfm Black 2 w(,)p 0 1 0 0 TeXcolorcmyk 442 w(A000931)p 45024 61058 V [[477 169 521 181] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000931) pdfm Black 2 w(,)j(and)p 0 1 0 0 TeXcolorcmyk 800 62450 a(A078027)p 800 62663 V [[79 155 123 167] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A078027) pdfm Black 436 w(.\))800 64856 y(Receiv)-36 b(ed)500 b(Jan)-36 b(uary)499 b(24,)517 b(2003;)533 b(revised)500 b(v)-36 b(ersion)499 b(receiv)-36 b(ed)500 b(July)f(2,)516 b(2003.)777 b(Published)498 b(in)h Fi(Journal)525 b(of)800 66461 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)431 b Fj(July)j(8,)g(2003.)p 800 68039 52000 45 v 800 70237 a(Return)f(to)p 0 1 0 0 TeXcolorcmyk 433 w(Journal)h(of)g(In)-36 b(teger)434 b(Sequences)f(home)g(page)p [[133 85 338 97] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterlo o.ca/JIS/) pdfm Black(.)p Black 26150 74617 a(12)p Black eop %%Trailer end end