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

1,5

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

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

Crossrefs

Cf. A002635, A295159.

Sequence in context: A291150 A292375 A361702 * A338430 A223135 A084113

Adjacent sequences: A295215 A295216 A295217 * A295219 A295220 A295221

Keyword

nonn

Author

Franklin T. Adams-Watters, Nov 17 2017

Status

approved