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A085409 Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.

%I A085409 #16 Jan 10 2023 01:51:44
%S A085409 0,12,84,270,624,1200,2052,3234,4800,6804,9300,12342,15984,20280,
%T A085409 25284,31050,37632,45084,53460,62814,73200,84672,97284,111090,126144,
%U A085409 142500,160212,179334,199920,222024,245700,271002,297984,326700,357204,389550,423792,459984
%N A085409 Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.
%C A085409 Parametric representation of the solution is (x, y, z) = (6n^3, 3n^3, 3n^2), thus getting a(n) = 9n^3 + 3n^2.
%H A085409 Colin Barker, <a href="/A085409/b085409.txt">Table of n, a(n) for n = 0..1000</a>
%H A085409 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A085409 a(n) = 9*n^3 + 3*n^2.
%F A085409 From _Colin Barker_, Oct 25 2019: (Start)
%F A085409 G.f.: 6*x*(2 + 6*x + x^2) /(1 - x)^4.
%F A085409 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
%F A085409 (End)
%F A085409 From _Amiram Eldar_, Jan 10 2023: (Start)
%F A085409 Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(3)*Pi/6 + 3*log(3)/2 - 3.
%F A085409 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 - Pi/sqrt(3) - 2*log(2) + 3. (End)
%t A085409 Table[9n^3 + 3n^2, {n, 0, 34}]
%o A085409 (PARI) concat(0, Vec(6*x*(2 + 6*x + x^2) /(1 - x)^4 + O(x^40))) \\ _Colin Barker_, Oct 25 2019
%o A085409 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 6*x*(2 + 6*x + x^2) /(1 - x)^4)); // _Marius A. Burtea_, Oct 25 2019
%Y A085409 Cf. A085377.
%K A085409 nonn,easy
%O A085409 0,2
%A A085409 Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 13 2003
%E A085409 More terms from _Robert G. Wilson v_, Aug 16 2003