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A263849 Let R = Z((1+sqrt{5})/2) denote the ring of integers in the real quadratic number field of discriminant 5. Let nu in R be a totally positive element of norm m = A031363(n). Then a(n) is the number of ways of writing nu as a sum of three squares in R. 3

%I #29 Nov 18 2015 12:11:38

%S 1,6,12,24,32,24,54,24,24,30,24,48,48,96,24,48,96,48,24,120,108,48,72,

%T 48,120,54,48,48,48,84,72,120,72,78,48,144

%N Let R = Z((1+sqrt{5})/2) denote the ring of integers in the real quadratic number field of discriminant 5. Let nu in R be a totally positive element of norm m = A031363(n). Then a(n) is the number of ways of writing nu as a sum of three squares in R.

%C Let R = Z((1+sqrt{5})/2) denote the ring of integers in the real quadratic number field of discriminant 5. The main result of Maass (1941) is that every totally positive nu in R is a sum of 3 squares x^2+y^2+z^2 with x,y,z in R. The number N_{nu} of such representations is given by the formula in the theorem on page 191. The norms of the totally positive elements nu are rational integers m belonging to A031363, so we can order the terms of the sequence according to the values m = A031363(n). [Comment based on remarks from Gabriele Nebe.]

%C The terms were computed with the aid of Magma by David Durstoff, Nov 11 2015.

%C The attached file from David Durstoff gives list of pairs m=A031363(n), a(n), and also the initial terms of Maass's series theta(tau). David Durstoff says: "I expressed theta(tau) in terms of two variables q1 and q2. The coefficient of q1^k q2^m is a(nu) with k = trace(nu/delta) and m = trace(nu), where delta = (5+sqrt{5})/2 is a generator of the different ideal. I computed the terms for q1^0 to q1^10 and all possible powers of q2."

%D Maass, Hans. Über die Darstellung total positiver Zahlen des Körpers R (sqrt(5)) als Summe von drei Quadraten, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 14. No. 1, pp. 185-191, 1941.

%H David Durstoff, <a href="/A263849/a263849.txt">Table showing list of pairs m=A031363(n), a(n) </a>

%Y Cf. A031363 (the norms), A035187 (number of ideals with that norm).

%Y See A263850 for another version of this sequence.

%K nonn,more

%O 0,2

%A _N. J. A. Sloane_, Nov 15 2015

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