A065459 -id:A065459 - 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 *)

A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 2, 6, 7, 6, 4, 8, 10, 14, 2, 11, 14, 13, 7, 23, 15, 17, 4, 24, 21, 31, 10, 25, 37, 28, 2, 46, 29, 49, 14, 38, 35, 61, 7, 45, 62, 49, 15, 93, 47, 57, 4, 72, 67, 97, 21, 71, 84, 101, 10, 119, 70, 86, 37, 92, 79, 165, 2, 138, 127, 109, 29, 168, 140, 121, 14

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=3 because 25 produces {0,0,0,5}, {0,0,3,4}, {1,2,2,4}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A063014, A016727, A065459.

Column k=4 of A255212.

N:= 100:
R:= Vector(N,1):
for a from 0 to N do
  for b from a to floor(sqrt(N^2-a^2)) do
     for c from b to floor(sqrt(N^2-a^2-b^2)) do
       q:= a^2 + b^2 + c^2;
       for f in numtheory:-divisors(q) do
          if f^2 + 2*f*c <= q and (f + q/f)::even then
             r:= (f + q/f)/2;
             if r <= N then R[r]:= R[r]+1 fi;
          fi
od od od od:
convert(R,list); # Robert Israel, Feb 16 2015

Length/@Table[SumOfSquaresRepresentations[4, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A179015 Number of ways in which n^2 can be expressed as the sum of exactly five positive squares.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 2, 6, 6, 9, 9, 15, 8, 25, 20, 21, 25, 39, 26, 46, 44, 57, 49, 71, 52, 102, 81, 81, 99, 145, 92, 156, 126, 164, 160, 204, 151, 247, 217, 236, 245, 326, 211, 357, 319, 381, 360, 416, 344, 518, 446, 476, 450, 670, 468, 675, 607, 661, 668, 825, 625

Offset: 1

Carmine Suriano, Jun 24 2010

nonn

As n goes to infinity the ratio of a(n)/a(n) of sequence A178898 (using all different squares) tends to 5/4.

Alois P. Heinz, Table of n, a(n) for n = 1..740

Cf. A000290, A178898.
Cf. A000132. - R. J. Mathar, Jun 26 2010
Cf. A065458, A065459.

a(8) = 5 since 64 can be expressed in five different ways as the sum of 5 squares (order is ignored): 8^2 = 7^2+3^2+2^2+1^2+1^2 = 6^2+5^2+1^2+1^2+1^2 = 6^2+4^2+2^2+2^2+2^2 = 6^2+3^2+3^2+3^2+1^2 = 5^2+5^2+3^2+1^2+1^2.

Asymptotic behavior for large values of n is a(n) = n^2/2-47n/2+243.
a(n) = A025429(n^2). - R. J. Mathar, Jun 26 2010
a(n) = A065459(n) - A065458(n). - Alois P. Heinz, Oct 25 2018

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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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A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

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A179015 Number of ways in which n^2 can be expressed as the sum of exactly five positive squares.

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