A267326 Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2).
1, 8, 24, 104, 24, 248, 312, 456, 24, 968, 744, 1064, 312, 1464, 1368, 3224, 24, 2456, 2904, 3048, 744, 5928, 3192, 4424, 312, 6248, 4392, 8744, 1368, 6968, 9672, 7944, 24, 13832, 7368, 14136, 2904, 11256, 9144, 19032, 744, 13784, 17784, 15144, 3192
Offset: 0
Examples
For n = 2 the a(n) = 24 solutions of x^2 + y^2 + z^2 + t^2 = 2^2 are:
{x,y,z,t} = {{0,0,0,2};{0,0,0,-2};{0,0,2,0};{0,0,-2,0};{0,2,0,0};{0,-2,0,0};{2,0,0,0};{-2,0,0,0};{1,1,1,1};{1,1,1,-1};{1,1,-1,1};{1,-1,1,1};{-1,1,1,1};{1,1,-1,-1};{1,-1,1,-1};{-1,1,1,-1};{1,-1,-1,1};{-1,1,-1,1};{1,-1,-1,-1};{-1,1,-1,-1};{-1,-1,1,-1};{-1,-1,1,-1};{-1,-1,-1,1};{-1,-1,-1,-1}}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..150 from Christopher Heiling)
Programs
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Maple
terms := 42: (add(q^(m^2), m = -terms..terms))^4: seq(coeff(%, q, n^2), n = 0..terms); # Peter Bala, Jan 15 2016
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Mathematica
a[n_] := SquaresR[4, n^2]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 18 2023 *)
Formula
a(n) = 8*sigma(n^2) if n is odd else 24*sigma(m(n^2)), where sigma(n) = A000203(n) and m(n) = A000265(n) is the largest odd divisor of n. - Peter Bala, Jan 15 2016
a(p^(k+1)) = 8*(p^2 *a(p^k)+p+1) for p prime. In particular a(p) = 8*(p^2+p+1). - Christopher Heiling, Apr 02 2017
Extensions
a(0)=1 prepended by Alois P. Heinz, Mar 10 2023
Comments