A295218 Number of partitions of 2*n-1 into four squares.

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 4, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 2, 5, 5, 4, 3, 6, 4, 5, 4, 5, 5, 5, 3, 6, 6, 5, 5, 6, 4, 5, 5, 5, 6, 8, 4, 6, 6, 7, 5, 7, 5, 7, 7, 6, 6, 6, 5, 8, 8, 6, 5, 10, 6, 8, 6, 7, 7, 8, 5, 8, 10, 7, 8, 8, 6, 8, 7, 9, 9, 11, 5, 8, 10, 7, 7

Offset: 1

Author

Franklin T. Adams-Watters, Nov 17 2017

Keywords

• nonn

Comments

This is a bisection of A002635.

While A002635 contains each positive integer infinitely often, here a number can appear only finitely many times.

By the Jacobi theorem, a(n) >= A000203(n)/48 >= (1+n)/48, which implies the previous comment. - Robert Israel, Nov 21 2017

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Jacobi four square theorem

Crossrefs

Cf. A002635, A295159.

Programs

Maple

N:= 100: # to get a(1)...a(N)
V:= Array(0..2*N-1):
for a from 0 while 4*a^2 <= 2*N-1 do
  for b from a while a^2 + 3*b^2 <= 2*N-1 do
     for c from b while a^2 + b^2 + 2*c^2 <= 2*N-1 do
       for d from c while a^2 + b^2 + c^2 + d^2 <= 2*N-1 do
         t:= a^2 + b^2 + c^2 + d^2;
         V[t]:= V[t]+1
od od od od:
seq(V[2*i-1],i=1..N); # Robert Israel, Nov 21 2017