A348165 Number of solutions to +-1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n.
1, 1, 0, 0, 1, 2, 0, 0, 2, 4, 0, 0, 19, 29, 0, 0, 127, 208, 0, 0, 1121, 1917, 0, 0, 10479, 19360, 0, 0, 113213, 204121, 0, 0, 1290968, 2363982, 0, 0, 15303057, 28397538, 0, 0, 187446097, 351339307, 0, 0, 2355979330, 4455357992, 0, 0, 30360404500, 57630025172
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..200
Programs
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Maple
b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1)+b(n+i^2, i-1))))((1+(3+2*i)*i)*i/6) end: a:= n-> `if`(irem(n, 4)>1, 0, b(n$2)): seq(a(n), n=0..50); # Alois P. Heinz, Jan 28 2022 -
Mathematica
b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[Abs[n - i^2], i - 1] + b[n + i^2, i - 1]]]][(1 + (3 + 2*i)*i)*i/6]; a[n_] := If[Mod[n, 4] > 1, 0, b[n, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *) -
Python
from functools import lru_cache @lru_cache(maxsize=None) def b(n, i): if n > i*(i+1)*(2*i+1)//6: return 0 if i == 0: return 1 return b(n+i**2, i-1) + b(abs(n-i**2), i-1) def a(n): return b(n, n) print([a(n) for n in range(50)]) # Michael S. Branicky, Jan 28 2022
Formula
a(n) = [x^n] Product_{k=1..n} (x^(k^2) + 1/x^(k^2)).