%I #9 Jul 23 2024 14:47:08
%S 593,595,623,637,697,707,733,833,965,1015,1037,1225,1295,1547,1585,
%T 1973,2023,2443,2597,3145,3227,3433,4165,5057,5383,5525,6713,7147,
%U 8687,8917,11245,11543,14035,14945,18173,18655,19865,24157,29377,31283,32113
%N Positive numbers y such that y^2 is of the form x^2+(x+833)^2 with integer x.
%C (-368, a(1)), (-357, a(2)), (-273, a(3)), (-245, a(4)), (-153, a(5)), (-140, a(6)), (-108, a(7)) and (A129010(n), a(n+7)) are solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.
%C lim_{n -> oo} a(n)/a(n-15) = 3+2*sqrt(2).
%C lim_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^4/((3+2*sqrt(2))*((19+6*sqrt(2))/17)^2) for n mod 15 = 1.
%C lim_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*((19+6*sqrt(2))/17)/((9+4*sqrt(2))/7)^3 for n mod 15 = {0, 2, 6, 11}.
%C lim_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 15 = {3, 5, 8, 9, 12, 14}.
%C lim_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 15 = {4, 7, 10, 13}.
%F a(n) = 6*a(n-15)-a(n-30) for n > 30.
%F G.f.: (1-x)*(593 +1188*x+1811*x^2+2448*x^3+3145*x^4+3852*x^5+4585*x^6+5418*x^7+6383*x^8+7398*x^9+8435*x^10+9660*x^11+10955*x^12+12502*x^13+14087*x^14+12502*x^15+10955*x^16+9660*x^17+8435*x^18+7398*x^19+6383*x^20+5418*x^21+4585*x^22+3852*x^23+3145*x^24+2448*x^25+1811*x^26+1188*x^27+593*x^28)/(1-6*x^15+x^30).
%e (-368, a(1)) = (-368, 593) is a solution: (-368)^2+(-368+833)^2 = 135424+216225 = 351649 = 593^2.
%e (A129010(1), a(8)) = (0, 833) is a solution: 0^2+(0+833)^2 = 693889 = 833^2.
%e (A129010(3), a(10)) = (168, 1015) is a solution: (168)^2+(168+833)^2 = 28224+1002001 = 1030225 = 1015^2.
%o (PARI) {forstep(n=-400, 26000, [3, 1], if(issquare(2*n^2+1666*n+693889, &k), print1(k, ",")))}
%Y Cf. A129010, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Feb 17 2009