A350205 -id:A350205 - cp's OEIS frontend

A279278 Expansion of Product_{k>=1} (1 + x^(k(k+1)(k+2)/6)).

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct tetrahedral numbers (A000292).

			a(35) = 2 because we have [35] and [20, 10, 4, 1].

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A000292, A007294, A024940, A068980, A350205 (positions of records).

nmax=120; CoefficientList[Series[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

A350210 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal pyramidal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

140, 490, 1055, 1872, 2610, 4255, 5011, 8708, 7497, 10819, 12860, 15636, 18055, 24275, 27373, 28146, 30826, 38178, 41849, 44025, 36165, 47621, 57896, 64648, 60064, 67125, 71975, 81820, 77701, 91584, 91320, 99835, 98916, 108686, 112606, 123180, 120919, 142270

Offset: 3

Ilya Gutkovskiy, Dec 19 2021

nonn

			For n = 3: 140 = 1 + 20 + 35 + 84 = 56 + 84 = 20 + 120. - _Martin Ehrenstein_, Jan 09 2022

Martin Ehrenstein, Table of n, a(n) for n = 3..1002
Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A080851, A350205, A350206, A350207, A350209, A350397.

a(35)-a(40) from Martin Ehrenstein, Jan 09 2022

A350206 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero square pyramidal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 91, 300, 490, 580, 525, 810, 910, 1100, 1050, 1330, 1340, 1430, 1400, 1625, 1750, 1540, 1700, 1820, 1955, 2010, 1995, 1980, 1925, 2185, 2280, 2190, 2275, 2360, 2535, 2445, 2415, 2505, 2500, 2605, 2645, 2640, 2785, 2780, 2865, 2870, 2925, 2800, 2835, 2940

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Cf. A000330, A097563, A298246, A350205.

A360214 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero octahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 231, 575, 721, 1618, 1750, 1877, 2240, 2736, 2995, 3105, 3080, 3500, 3311, 3920, 4151, 4280, 4495, 4719, 4621, 4675, 5041, 5164, 5291, 5060, 5591, 5480, 5566, 5635, 5755, 5985, 6216, 6080, 6279, 6320, 6510, 6655, 6636, 6870, 7145, 7195, 6999, 6971, 7296, 7211

Offset: 1

Ilya Gutkovskiy, Jan 30 2023

nonn

Eric Weisstein's World of Mathematics, Octahedral Number

Cf. A005900, A275154, A350205, A360215, A360216.

A360215 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero icosahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 1383, 4157, 6548, 8633, 9884, 12503, 12920, 15357, 15812, 18146, 18126, 19755, 20895, 22106, 23229, 23246, 23685, 22118, 25142, 25884, 27894, 29448, 28149, 29703, 30285, 31914, 30966, 34007, 34380, 34390, 35082, 35894, 34389, 36891, 37035, 37425, 35907, 35895, 38856

Offset: 1

Ilya Gutkovskiy, Jan 30 2023

nonn

Wikipedia, Icosahedral number

Cf. A006564, A275154, A350205, A360214, A360216.

A360216 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero dodecahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 2025, 2925, 9010, 15521, 18465, 19140, 24899, 32760, 33576, 36245, 39746, 39290, 39270, 46540, 50215, 49055, 53680, 50435, 56585, 58990, 57460, 58380, 61950, 63329, 64600, 63700, 64550, 67305, 68530, 71690

Offset: 1

Ilya Gutkovskiy, Jan 30 2023

nonn

Wikipedia, Dodecahedral number

Cf. A006566, A275154, A350205, A360214, A360215.

A360217 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero tetrahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 140, 305, 315, 435, 644, 830, 1141, 1425, 1925, 2380, 3010, 3805, 4720, 5806, 7095, 8510, 10200, 12020, 14115, 16460, 19131, 21990, 25425, 29275, 33495, 37425, 42680, 48300, 54545, 60711, 68391, 75726, 84815, 93370, 103250, 114115, 125360, 137831, 150995, 165545, 179830

Offset: 1

Ilya Gutkovskiy, Jan 30 2023

nonn

			For n = 3: 305 = 1 + 84 + 220 = 20 + 120 + 165 = 56 + 84 + 165.

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A350205, A350288, A350397, A360218.

More terms from David A. Corneth, Jan 30 2023

cp's OEIS Frontend

A279278 Expansion of Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

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A350210 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal pyramidal numbers in exactly n ways, or 0 if no such integer exists.

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A350206 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero square pyramidal numbers in exactly n ways, or 0 if no such integer exists.

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A360214 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero octahedral numbers in exactly n ways, or -1 if no such integer exists.

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A360215 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero icosahedral numbers in exactly n ways, or -1 if no such integer exists.

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A360216 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero dodecahedral numbers in exactly n ways, or -1 if no such integer exists.

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A360217 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero tetrahedral numbers in exactly n ways, or -1 if no such integer exists.

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A279278 Expansion of Product_{k>=1} (1 + x^(k(k+1)(k+2)/6)).