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A159690 Positive numbers y such that y^2 is of the form x^2+(x+881)^2 with integer x.

%I A159690 #10 Sep 08 2022 08:45:44
%S A159690 841,881,925,4121,4405,4709,23885,25549,27329,139189,148889,159265,
%T A159690 811249,867785,928261,4728305,5057821,5410301,27558581,29479141,
%U A159690 31533545,160623181,171817025,183790969,936180505,1001423009,1071212269
%N A159690 Positive numbers y such that y^2 is of the form x^2+(x+881)^2 with integer x.
%C A159690 (-41,a(1)) and (A130014(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.
%C A159690 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A159690 lim_{n -> infinity} a(n)/a(n-1) = (883+42*sqrt(2))/881 for n mod 3 = {0, 2}.
%C A159690 lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 1.
%H A159690 G. C. Greubel, <a href="/A159690/b159690.txt">Table of n, a(n) for n = 1..3501</a>
%H A159690 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A159690 a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=841, a(2)=881, a(3)=925, a(4)=4121, a(5)=4405, a(6)=4709.
%F A159690 G.f.: (1-x)*(841+1722*x+2647*x^2+1722*x^3+841*x^4) / (1-6*x^3+x^6).
%F A159690 a(3*k-1) = 881*A001653(k) for k >= 1.
%e A159690 (-41, a(1)) = (-41, 841) is a solution: (-41)^2+(-41+881)^2 = 1681+705600 = 707281 = 841^2.
%e A159690 (A130014(1), a(2)) = (0, 881) is a solution: 0^2+(0+881)^2 = 776161 = 881^2.
%e A159690 (A130014(3), a(4)) = (2440, 4121) is a solution: 2440^2+(2440+881)^2 = 5953600+11029041 = 16982641 = 4121^2.
%t A159690 CoefficientList[Series[(1 - x)*(841 + 1722*x + 2647*x^2 + 1722*x^3 + 841*x^4)/(1 - 6*x^3 + x^6), {x,0,50}], x] (* or *) LinearRecurrence[{0, 0,6,0,0,-1}, {841, 881, 925, 4121, 4405, 4709}, 30] (* _G. C. Greubel_, Jun 02 2018 *)
%o A159690 (PARI) {forstep(n=-44, 10000000, [3, 1], if(issquare(2*n^2+1762*n+776161, &k), print1(k, ",")))}
%o A159690 (Magma) I:=[841, 881, 925, 4121, 4405, 4709]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, Jun 02 2018
%Y A159690 Cf. A130014, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).
%K A159690 nonn,easy
%O A159690 1,1
%A A159690 _Klaus Brockhaus_, Apr 21 2009