A065461 -id:A065461 - 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 *)

A361695 Number of ways of writing n^2 as a sum of seven squares.

Original entry on oeis.org

1, 14, 574, 3542, 18494, 43414, 145222, 235998, 591934, 860846, 1779974, 2256422, 4678982, 5195750, 9675918, 10983742, 18942014, 19873966, 35294686, 34670454, 57349894, 59707494, 92513302, 90116222, 149759302, 135668414, 213025750, 209185718, 311753358, 287144326, 450333422

Offset: 0

Alois P. Heinz, Mar 22 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to sums of squares

Column k=7 of A302996.

Cf. A008451, A065461.

a:= n-> coeff((sum(x^(j^2), j=-n..n))^7, x, n^2):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 7):
seq(a(n), n=0..30);

SquaresR[7, Range[0, 30]^2] (* Paolo Xausa, Aug 21 2025 *)

a(n) = [x^(n^2)] (Sum_{j=-n..n} x^(j^2))^7.
a(n) = A008451(n^2).
a(n) = A302996(n,7).

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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