Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #22 Dec 14 2018 07:52:49
%S 1,2,1,2,4,4,3,4,5,6,8,6,5,10,7,8,14,8,9,12,11,12,12,12,14,18,13,12,
%T 22,16,15,24,12,18,24,18,19,20,24,20,28,22,16,28,23,24,32,20,25,38
%N Number of ordered integral solutions of Descartes' equation 2(a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2 with a + b + c + d = 4n - 2.
%C This count includes only primitive quadruples, i.e., with gcd=1. There may also be nonprimitive quads, e.g., with n=5, 4n-2=18, we have 4 primitive quads (-1,2,6,11), (-2,3,7,10), (1,1,4,12), (-3,5,8,8) and also the nonprimitive (0,3,3,12) and (3,6,6,9). - _Colin Mallows_, May 11 2005
%H R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, <a href="http://arxiv.org/abs/math/0009113">Apollonian Circle Packings: Number Theory</a>, arXiv:math/0009113 [math.NT], 2000-2003; J. Number Theory, 100 (2003), 1-45.
%H J. C. Lagarias, C. L. Mallows, and Allan Wilks, <a href="http://arxiv.org/abs/math/0101066">Beyond the Descartes Circle Theorem</a>, arXiv:math/0101066 [math.MG], 2001.
%H J. C. Lagarias, C. L. Mallows, and Allan Wilks, <a href="http://www.jstor.org/stable/2695498">Beyond the Descartes Circle Theorem</a>, Amer. Math Monthly, 109 (2002), 338-361.
%e a(5) = 4 because we have the quadruples (1,1,4,12), (-1,2,6,11), (-2,3,7,10), (3,5,8,8).
%t r[n_] := Reduce[a <= b <= c <= d && 2 (a^2 + b^2 + c^2 + d^2) == (a + b + c + d)^2 && a + b + c + d == 4 n - 2, {a, b, c, d}, Integers];
%t a[n_] := Count[{a, b, c, d} /. {ToRules[r[n]]}, sol_ /; GCD @@ sol == 1];
%t Reap[Do[an = a[n]; Print[n, " ", an]; Sow[an], {n, 1, 50}]][[2, 1]] (* _Jean-François Alcover_, Dec 14 2018 *)
%Y Cf. A045864.
%K nonn
%O 1,2
%A _Colin Mallows_, Apr 28 2005
%E More terms from _Colin Mallows_, May 11 2005