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A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

%I A129642 #17 Aug 31 2025 19:21:53
%S A129642 0,348,495,1371,3255,4088,9140,20096,24947,54383,118235,146508,318072,
%T A129642 690228,855015,1854963,4024047,4984496,10812620,23454968,29052875,
%U A129642 63021671,136706675,169333668,367318320,796785996,986950047,2140889163,4644010215,5752367528
%N A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.
%C A129642 Also values x of Pythagorean triples (x, x+457, y).
%C A129642 Corresponding values y of solutions (x, y) are in A160580.
%C A129642 Limit_{n->oo} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129642 Limit_{n->oo} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {1, 2}.
%C A129642 Limit_{n->oo} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 0.
%H A129642 Harvey P. Dale, <a href="/A129642/b129642.txt">Table of n, a(n) for n = 1..1000</a>
%H A129642 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129642 a(n) = 6*a(n-3)-a(n-6)+914 for n > 6; a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088.
%F A129642 G.f.: x^2*(348+147*x+876*x^2-204*x^3-49*x^4-204*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A129642 a(3*k+1) = 457*A001652(k) for k >= 0.
%F A129642 a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088, a(7)=9140, a(n) = a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - _Harvey P. Dale_, May 13 2012
%t A129642 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,348,495,1371,3255,4088,9140},30] (* _Harvey P. Dale_, May 13 2012 *)
%o A129642 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849), print1(n, ",")))}
%Y A129642 Cf. A160580, A001652, A129641, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).
%K A129642 nonn,easy,changed
%O A129642 1,2
%A A129642 _Mohamed Bouhamida_, May 31 2007
%E A129642 Edited and two terms added by _Klaus Brockhaus_, Jun 08 2009