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

0,2

Comments

For all pair of relatively prime numbers k, m this sequence is multiplicative with a factor of 8: a(k*m) = 8*a(k)*a(m). - Christopher Heiling, Apr 02 2017

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..150 from Christopher Heiling)

Formula

a(n) = A264390(n) - A264390(n-1) for n > 1 and a(1) = A264390(1) = 2*D.

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

Example

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}}.

Maple

terms := 42:
(add(q^(m^2), m = -terms..terms))^4:
seq(coeff(%, q, n^2), n = 0..terms); # Peter Bala, Jan 15 2016

Mathematica

a[n_] := SquaresR[4, n^2];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 18 2023 *)

Crossrefs

Cf. A000118.

Partial sums of this sequence give A264390.

Column k=4 of A302996.

Sequence in context: A019221 A334756 A281463 * A063515 A220706 A246030

Adjacent sequences: A267323 A267324 A267325 * A267327 A267328 A267329

Keyword

nonn,easy

Author

Christopher Heiling, Jan 13 2016

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 10 2023

Status

approved