A160212 Positive numbers y such that y^2 is of the form x^2+(x+953)^2 with integer x.

845, 953, 1093, 3977, 4765, 5713, 23017, 27637, 33185, 134125, 161057, 193397, 781733, 938705, 1127197, 4556273, 5471173, 6569785, 26555905, 31888333, 38291513, 154779157, 185858825, 223179293, 902119037, 1083264617, 1300784245

Offset

1,1

Comments

(-116, a(1)) and (A129975(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 1.

Links

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=845, a(2)=953, a(3)=1093, a(4)=3977, a(5)=4765, a(6)=5713.

G.f.: (1-x)*(845+1798*x+2891*x^2+1798*x^3+845*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 953*A001653(k) for k >= 1.

Example

(-116, a(1)) = (-116, 845) is a solution: (-116)^2+(-116+953)^2 = 13456+700569 = 714025 = 845^2.
(A129975(1), a(2)) = (0, 953) is a solution: 0^2+(0+953)^2 = 908209 = 953^2.
(A129975(3), a(4)) = (2295, 3977) is a solution: 2295^2+(2295+953)^2 = 5267025+10549504 = 15816529 = 3977^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {845, 953, 1093, 3977, 4765, 5713}, 30] (* Harvey P. Dale, Feb 18 2024 *)

Prog

(PARI) {forstep(n=-116, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209, &k), print1(k, ", ")))}

Crossrefs

Cf. A129975, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

Sequence in context: A071320 A338628 A323253 * A188296 A187861 A280484

Adjacent sequences: A160209 A160210 A160211 * A160213 A160214 A160215

Keyword

nonn

Author

Klaus Brockhaus, May 18 2009

Status

approved