A025429 Number of partitions of n into 5 nonzero squares.

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 0, 1, 3, 1, 3, 2, 1, 3, 2, 1, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 5, 2, 2, 5, 1, 3, 5, 1, 5, 4, 2, 5, 3, 2, 5, 5, 3, 4, 4, 4, 3, 5, 4, 4, 7, 3, 5, 6, 2, 4, 7, 4, 7, 6, 3, 7, 4, 3, 8, 6, 5, 7, 5, 5, 4, 6, 7, 6, 9, 5, 6, 8, 2, 8

Offset

0,21

Comments

a(33) is the last zero in this sequence, cf. the link to Mathematics Stack Exchange and also A080673(n) for the largest index k with a(k)=n. - M. F. Hasler, May 30 2014

First occurrence of k beginning with 0: 0, 5, 20, 29, 62, 53, 80, 77, 91, 101, ..., (A080654). - Robert G. Wilson v, May 30 2014

Links

M. F. Hasler, Table of n, a(n) for n = 0..10000

H. v. Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on Mathematics Stack Exchange, May 2014

Index entries for sequences related to sums of squares

Formula

a(n) = [x^n y^5] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(n-i-j-k-l). - Wesley Ivan Hurt, Apr 19 2019

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 5):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

Mathematica

f[n_] := Block[{c = Range@ Sqrt@ n^2}, Length@ IntegerPartitions[n, {5}, c]]; Array[f, 105, 0] (* Robert G. Wilson v, May 30 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[i<1 || t<1, 0, b[n, i-1, t] + If[i^2>n, 0, b[n-i^2, i, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 5]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

Prog

(PARI)  A025429(n)=sum(d=sqrtint(max(n, 5)\5), sqrtint(max(n-4, 0)), nn=n-d^2; sum(a=sqrtint(max(nn-d^2, 4)\4), min(sqrtint(max(nn-3, 0)), d), sum(b=sqrtint((nn-a^2)\3-1)+1, min(sqrtint(nn-a^2-2), a), sum(c=sqrtint((t=nn-a^2-b^2)\2-1)+1, min(sqrtint(t-1), b), issquare(t-c^2) )))) \\ M. F. Hasler, May 30 2014

Crossrefs

Column k=5 of A243148.

Sequence in context: A098744 A337584 A273975 * A325561 A076250 A360963

Adjacent sequences: A025426 A025427 A025428 * A025430 A025431 A025432

Keyword

nonn,easy,look

Author

David W. Wilson

Status

approved