A105152 -id:A105152 - cp's OEIS frontend

A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 2, 3, 3, 3, 2, 1, 1, 0, 1, 1, 2, 3, 3, 4, 4, 2, 1, 1, 0, 1, 1, 2, 3, 4, 5, 5, 4, 1, 1, 1, 0, 1, 1, 2, 4, 5, 7, 9, 6, 2, 4, 2, 1, 0

Offset: 0

Alois P. Heinz, Feb 17 2015

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 2,  2,  2,  2,  2,  2,  2, ...
  0, 1, 1, 2, 2,  2,  3,  3,  3,  4,  4, ...
  0, 1, 1, 1, 2,  3,  3,  4,  5,  5,  6, ...
  0, 1, 2, 2, 3,  4,  5,  7,  8,  9, 11, ...
  0, 1, 1, 2, 4,  5,  9, 10, 11, 15, 17, ...
  0, 1, 1, 2, 4,  6,  9, 13, 18, 21, 27, ...
  0, 1, 1, 1, 2,  7,  9, 16, 25, 30, 41, ...
  0, 1, 1, 4, 6,  8, 18, 27, 36, 52, 68, ...
  0, 1, 2, 2, 7, 13, 23, 36, 51, 70, 94, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000012, A063014, A016727, A065458, A065459, A065460, A065461, A065462, A255213, A255214.
Main diagonal gives A105152.
Cf. A302996.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1 and n<=t, 1,
      (j-> `if`(t*jn, 0, b(n-j, i, t-1))))(i^2))
    end:
A:= (n, k)-> b(n^2, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1 && n <= t, 1, Function[j, If[t*jn, 0, b[n-j, i, t-1]]]][i^2]]; A[n_, k_] := b[n^2, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

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

Original entry on oeis.org

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

A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 7, 1, 31, 123, 151, 121, 897, 7351, 5415, 14881, 48705, 150583, 468973, 1013163, 1432471, 1730023, 50432107, 14925241, 125269841, 74592537, 241763479, 213156871, 895153173, 7716880623, 2681163865, 3190865761, 22501985413, 116279718801

Offset: 0

Ilya Gutkovskiy, Jan 30 2020

nonn

			a(5) = 3 because we have [25], [16, 9] and [9, 16].

Cf. A000290, A006456, A030273, A032020, A037444, A105152, A224366, A232173, A280129, A298640, A331844.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^2, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^2, n, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

a(n) = A331844(A000290(n)).

a(24)-a(34) from Alois P. Heinz, Jan 30 2020

A307739 Number of partitions of n^4 into at most n fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 5, 3, 5, 2, 27, 4, 78, 14, 152, 551, 1331, 7377, 15504, 87583, 190028, 768864, 1510305, 5413291, 12221733

Offset: 0

Ilya Gutkovskiy, Apr 25 2019

			10^4 =
4^4 + 4^4 + 6^4 + 8^4 + 8^4 =
2^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 8^4 + 8^4,
so a(10) = 3.

Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A105152, A259793, A299169, A299195, A307644, A307738.

from sympy.solvers.diophantine.diophantine import power_representation
def a(n):
    if n < 2: return 1
    return sum(len(list(power_representation(n**4, 4, j))) for j in range(1, n+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jul 09 2024

a(21)-a(27) from Michael S. Branicky, Jul 09 2024

A347590 Number of compositions (ordered partitions) of n^2 into at most n squares.

Original entry on oeis.org

1, 1, 1, 4, 2, 20, 188, 1031, 8777, 62528, 437160, 4185739, 38642386, 383969125, 4149154916, 45160025119, 514181220266, 6133093344169, 75135177511922, 962729735639323, 12745694628358530, 173304634479902187, 2433186864257121180, 35030916937968941062

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A006456, A105152, A224366, A298330, A347591, A347592.

cp's OEIS Frontend

A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

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A307739 Number of partitions of n^4 into at most n fourth powers.

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A347590 Number of compositions (ordered partitions) of n^2 into at most n squares.

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