A156835 Positive numbers y such that y^2 is of the form x^2+(x+833)^2 with integer x.

593, 595, 623, 637, 697, 707, 733, 833, 965, 1015, 1037, 1225, 1295, 1547, 1585, 1973, 2023, 2443, 2597, 3145, 3227, 3433, 4165, 5057, 5383, 5525, 6713, 7147, 8687, 8917, 11245, 11543, 14035, 14945, 18173, 18655, 19865, 24157, 29377, 31283, 32113

Offset: 1

Author

Klaus Brockhaus, Feb 17 2009

Keywords

• nonn

Comments

(-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.

lim_{n -> oo} a(n)/a(n-15) = 3+2*sqrt(2).

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.

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}.

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}.

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}.

Examples

			(-368, a(1)) = (-368, 593) is a solution: (-368)^2+(-368+833)^2 = 135424+216225 = 351649 = 593^2.
(A129010(1), a(8)) = (0, 833) is a solution: 0^2+(0+833)^2 = 693889 = 833^2.
(A129010(3), a(10)) = (168, 1015) is a solution: (168)^2+(168+833)^2 = 28224+1002001 = 1030225 = 1015^2.
		

Crossrefs

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).

Programs

PARI

{forstep(n=-400, 26000, [3, 1], if(issquare(2*n^2+1666*n+693889, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-15)-a(n-30) for n > 30.

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).