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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
1, 6, 12, 24, 32, 24, 54, 24, 24, 30, 24, 48, 48, 96, 24, 48, 96, 48, 24, 120, 108, 48, 72, 48, 120, 54, 48, 48, 48, 84, 72, 120, 72, 78, 48, 144 (list; graph; refs; listen; history; text; internal format)



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.]

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

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."


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.


Table of n, a(n) for n=0..35.

David Durstoff, Table showing list of pairs m=A031363(n), a(n)  


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

See A263850 for another version of this sequence.

Sequence in context: A260633 A110967 A000082 * A227416 A106697 A323002

Adjacent sequences:  A263846 A263847 A263848 * A263850 A263851 A263852




N. J. A. Sloane, Nov 15 2015



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Last modified July 18 21:25 EDT 2019. Contains 325144 sequences. (Running on oeis4.)