A307643 -id:A307643 - cp's OEIS frontend

A307738 Number of partitions of n^3 into at most n cubes.

1, 1, 1, 1, 1, 1, 3, 4, 7, 18, 36, 66, 157, 329, 728, 1611, 3655, 8062, 18154, 40358, 89807, 199778, 444419, 984422, 2183461, 4827756, 10651083, 23465459, 51576034, 113092423, 247546849, 540538832, 1177836149, 2560897979, 5555722749, 12025952101, 25976048200

Offset: 0

Ilya Gutkovskiy, Apr 25 2019

nonn

Does a(n+1) / a(n) ~ 2? - David A. Corneth, Sep 27 2019

			7^3 =
1^3 + 1^3 + 5^3 + 6^3 =
1^3 + 1^3 + 3^3 + 4^3 + 5^3 + 5^3 =
1^3 + 2^3 + 3^3 + 3^3 + 4^3 + 6^3,
so a(7) = 4.

David A. Corneth, Table of n, a(n) for n = 0..100
Index entries for sequences related to sums of cubes

Cf. A000578, A025446-A025454, A030272, A105152, A259792, A298671, A298672, A307643, A307739.

a(n) = {my(res = 0); res=aIterate(n^3, 1, n); res }
aIterate(s, m, q) = { if(s == 0, return(1)); if(q == 0, return(0)); sum(i = m, sqrtnint(s, 3), aIterate(s - i^3, i, q-1) ) } \\ David A. Corneth, Sep 23 2019

a(21)-a(36) from David A. Corneth, Sep 23 2019

A338586 Number of partitions of the n-th tetrahedral number into exactly n positive tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 5, 5, 20, 35, 75, 154, 336, 730, 1570, 3394, 7339, 16085, 35015, 76269, 164821, 359704, 782004, 1696804, 3668860, 7953962, 17184203, 37093184, 79825297, 171824175, 368838299, 790404448, 1690297309, 3610816466, 7696144659, 16374004711, 34766160358

Offset: 0

Ilya Gutkovskiy, Nov 08 2020

nonn

			The 6th tetrahedral number is 56 and 56 = 1 + 1 + 4 + 10 + 20 + 20 = 4 + 4 + 4 + 4 + 20 + 20, so a(6) = 2.

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

Cf. A000292, A068980, A298269, A298857, A303170, A307643, A338585, A338778.

a(n) = [x^A000292(n) y^n] Product_{j>=1} 1 / (1 - y*x^A000292(j)).

A307644 Number of partitions of n^4 into exactly n nonzero fourth powers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 6, 0, 27, 13, 59, 390, 661, 4933, 9760, 49415, 101967, 341887, 702884, 2209559, 5361004, 15472531, 34165997, 82258594, 193682533, 490404772, 1210929426, 2725005202, 6283337761, 13672859806, 34906926846

Offset: 0

Ilya Gutkovskiy, Apr 19 2019

nonn

			11^4 =
1^4 + 2^4 + 2^4 + 4^4 + 4^4 + 4^4 + 4^4 + 6^4 + 8^4 + 8^4 + 8^4 =
2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 6^4 + 6^4 + 6^4 + 8^4 + 9^4 =
2^4 + 2^4 + 2^4 + 4^4 + 6^4 + 6^4 + 6^4 + 6^4 + 6^4 + 6^4 + 9^4,
so a(11) = 3.

Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A259793, A299169, A299195, A307643, A319435.

a(20)-a(28) from Vaclav Kotesovec, Apr 20 2019

a(29)-a(37) from Vaclav Kotesovec, Apr 23 2019

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A307738 Number of partitions of n^3 into at most n cubes.

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A338586 Number of partitions of the n-th tetrahedral number into exactly n positive tetrahedral numbers.

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A307644 Number of partitions of n^4 into exactly n nonzero fourth powers.

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