A097773 Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n) = A097772(n), n >= 0.

1, 677, 459005, 311204713, 210996336409, 143055204880589, 96991217912702933, 65759902689607707985, 44585117032336113310897, 30228643588021195217080181, 20494975767561338021067051821, 13895563341762999157088244054457, 9421171450739545867167808401870025

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..353

Tanya Khovanova, Recursive Sequences.

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum, Vol. 19 (2019), 11-16.

Index entries for linear recurrences with constant coefficients, signature (678,-1).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = ((-1)^n)*S(2*n, 26*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-678*x+x^2).

a(n) = S(n, 2*339) - S(n-1, 2*339) = T(2*n+1, sqrt(170))/sqrt(170), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

a(n) = 678*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=677. - Philippe Deléham, Nov 18 2008

Sum_{n>=0} 1/(a(n)+1) = sqrt(170)/26. - Amiram Eldar, Jan 01 2026

Example

(x, y) = (13*1 = 13, 1), (8827 = 13*679, 677), (5984693 = 13*460361, 459005), ... give the positive integer solutions to x^2 - 170*y^2 = -1.

Mathematica

LinearRecurrence[{678, -1}, {1, 677}, 11] (* Ray Chandler, Aug 12 2015 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-678*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 677]; [n le 2 select I[n] else 678*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(SageMath) ((1-x)/(1-678*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 677];; for n in [3..20] do a[n]:=678*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Cf. A097771 for S(n, 678).

Row 13 of array A188647.

Cf. A049310, A053120, A097772.

Sequence in context: A108824 A205471 A144381 * A248887 A031524 A158385

Adjacent sequences: A097770 A097771 A097772 * A097774 A097775 A097776

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved