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

Author

Klaus Brockhaus, May 18 2009

Keywords

• nonn

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.

Examples

			(-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.
		

Links

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

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).

Programs

Mathematica

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

PARI

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

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.