A288342 -id:A288342 - cp's OEIS frontend

A288254 Number of octagons that can be formed with perimeter n.

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 104, 134, 167, 211, 258, 322, 389, 480, 572, 698, 825, 996, 1165, 1395, 1620, 1923, 2216, 2611, 2991, 3500, 3984, 4633, 5248, 6066, 6836, 7860, 8820, 10089, 11273, 12835, 14288, 16197

Offset: 8

Seiichi Manyama, Jun 07 2017

Number of (a1, a2, ... , a8) where 1 <= a1 <= ... <= a8 and a1 + a2 + ... + a7 > a8.

Seiichi Manyama, Table of n, a(n) for n = 8..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 1, -1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 2, -2, 0, 0, 0, 0, 0, 0, -2, 2, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, -1, 1, 1, -1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), this sequence (k=8), A288255 (k=9), A288256 (k=10).

G.f.: x^8/((1-x)*(1-x^2)* ... *(1-x^8)) - x^14/(1-x) * 1/((1-x^2)*(1-x^4)* ... *(1-x^14)).

a(2*n+14) = A026814(2*n+14) - A288342(n), a(2*n+15) = A026814(2*n+15) - A288342(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A288341 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^6)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 44, 64, 90, 125, 169, 227, 298, 388, 498, 634, 797, 996, 1231, 1513, 1844, 2235, 2689, 3221, 3833, 4542, 5353, 6284, 7341, 8547, 9907, 11447, 13176, 15121, 17293, 19725, 22427, 25436, 28767, 32459, 36529, 41023, 45958, 51385, 57327

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 6 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -2, 2, 1, 0, 0, 0, -1, -2, 2, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), this sequence (k=6), A288342 (k=7), A288343 (k=8), A288344 (k=9), A288345 (k=10).

Cf. A288253. Column 6 of A092905. A001402 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 6, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A288344 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^9)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 138, 192, 265, 359, 482, 639, 840, 1092, 1410, 1803, 2291, 2889, 3621, 4508, 5584, 6875, 8424, 10269, 12463, 15055, 18115, 21704, 25910, 30814, 36522, 43137, 50794, 59618, 69774, 81422, 94760, 109984, 127338, 147058, 169438

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 9 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, 0, -1, 1, 2, -1, 0, 0, -1, -1, 0, 0, -1, 1, 0, 2, 0, 1, -1, 0, 0, -1, -1, 0, 0, -1, 2, 1, -1, 0, 1, -1, 1, -1, 0, -1, 0, 2, -1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), A288343 (k=8), this sequence (k=9), A288345 (k=10).

Cf. A288256, A008638 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 9, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A288345 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^10)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 269, 366, 494, 658, 870, 1137, 1477, 1900, 2430, 3083, 3890, 4874, 6078, 7533, 9294, 11406, 13940, 16955, 20545, 24787, 29800, 35688, 42600, 50670, 60088, 71024, 83714, 98377, 115305, 134771, 157138, 182746, 212038

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 10 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, 0, 0, -1, 2, 0, 0, 1, -2, 0, -1, 0, 0, 0, -2, 3, 0, 1, 0, 1, 0, -1, 0, -1, 0, -3, 2, 0, 0, 0, 1, 0, 2, -1, 0, 0, -2, 1, 0, 0, -1, 1, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), A288343 (k=8), A288344 (k=9), this sequence (k=10).

Cf. A008639 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 10, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A288343 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^8)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 96, 136, 188, 258, 347, 463, 609, 795, 1025, 1313, 1665, 2099, 2624, 3262, 4026, 4945, 6035, 7332, 8859, 10660, 12764, 15226, 18083, 21402, 25230, 29647, 34713, 40525, 47155, 54719, 63307, 73056, 84074, 96524, 110536, 126301

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 8 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -2, 0, -1, 1, 0, 2, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), this sequence (k=8), A288344 (k=9), A288345 (k=10).

Cf. A288255.

CoefficientList[Series[1/((1-x)Times@@(1-x^Range[8])),{x,0,50}],x] (* Harvey P. Dale, Dec 06 2017 *)

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 8, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

cp's OEIS Frontend

A288254 Number of octagons that can be formed with perimeter n.

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Formula

A288341 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^6)).

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A288344 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^9)).

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A288345 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^10)).

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A288343 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^8)).

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PARI

A288341 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^6)).

A288344 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^9)).

A288345 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^10)).

A288343 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^8)).