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A267326 Number of ways writing n^2 as a sum of four squares: a(n) = A000118(n^2). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Christopher Heiling, Table of n, a(n) for n = 1..150

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 is 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

#A267326

terms := 42:

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

seq(coeff(%, q, n^2), n = 1..terms); # Peter Bala, Jan 15 2016

CROSSREFS

Cf. A000118.

Partial sums of this sequence give A264390.

Sequence in context: A019260 A019221 A281463 * A063515 A220706 A246030

Adjacent sequences:  A267323 A267324 A267325 * A267327 A267328 A267329

KEYWORD

nonn,easy

AUTHOR

Christopher Heiling, Jan 13 2016

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)