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%I #23 Jul 03 2023 08:26:58
%S 1,40,1089,25160,531521,10625640,204744769,3844391560,70827391041,
%T 1286290883240,23101397290049,411249127989960,7269184506192961,
%U 127745926316548840,2234231991096868929,38920247688751940360
%N Sum_(x^i*y^j*z^k) with i + j + k = m and (x, y, z) = the primitive Pythagorean triple (8, 15, 17).
%D G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 196.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (40, -511, 2040).
%F a(m) = (x^(m+2)*(z-y) + y^(m+2)*(x-z) + z^(m+2)*(y-x))/((x-y)*(y-z)*(z-x)).
%F From _Chai Wah Wu_, Sep 24 2016: (Start)
%F a(n) = 40*a(n-1) - 511*a(n-2) + 2040*a(n-3) for n > 2.
%F G.f.: 1/((1 - 8*x)*(1 - 15*x)*(1 - 17*x)). (End)
%F a(n) = 2^(3*n+6)/63 - 15^(n+2)/14 + 17^(n+2)/18. - _Vaclav Kotesovec_, Sep 25 2016
%e a(2) = 1089 because x^2 + y^2 + z^2 + x*y + x*z + y*z = 8^2 + 15^2 + 17^2 + 8*15 + 8*17 + 15*17 = 1089 and x^2 + y^2 = z^2.
%p seq(sum(8^(m-n)*sum(15^p*17^(n-p),p=0..n),n=0..m),m=0..N);
%Y Cf. A019682, A020000, A020340-A020342, A020344-A020346, A021664, A021684, A021844, A025942, A077515.
%K nonn
%O 0,2
%A _Giorgio Balzarotti_ and _Paolo P. Lava_, Nov 26 2006
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