A267326 - OEIS

%I #66 May 18 2023 08:33:04

%S 1,8,24,104,24,248,312,456,24,968,744,1064,312,1464,1368,3224,24,2456,

%T 2904,3048,744,5928,3192,4424,312,6248,4392,8744,1368,6968,9672,7944,

%U 24,13832,7368,14136,2904,11256,9144,19032,744,13784,17784,15144,3192

%N Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2).

%C 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

%H Alois P. Heinz, <a href="/A267326/b267326.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..150 from Christopher Heiling)

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

%F 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

%F 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

%e For n = 2 the a(n) = 24 solutions of x^2 + y^2 + z^2 + t^2 = 2^2 are:

%e {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}}.

%p terms := 42:

%p (add(q^(m^2), m = -terms..terms))^4:

%p seq(coeff(%, q, n^2), n = 0..terms); # _Peter Bala_, Jan 15 2016

%t a[n_] := SquaresR[4, n^2];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, May 18 2023 *)

%Y Cf. A000118.

%Y Partial sums of this sequence give A264390.

%Y Column k=4 of A302996.

%K nonn,easy

%O 0,2

%A _Christopher Heiling_, Jan 13 2016

%E a(0)=1 prepended by _Alois P. Heinz_, Mar 10 2023