A097767 Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n) = A097766(n), n >= 0.

1, 485, 235709, 114554089, 55673051545, 27056988496781, 13149640736384021, 6390698340894137425, 3105866244033814404529, 1509444603902092906463669, 733586971630173118726938605, 356521758767660233608385698361, 173268841174111243360556722464841, 84208300288859296612996958732214365

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..372

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 (486,-1).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = S(n, 2*243) - S(n-1, 2*243) = T(2*n+1, sqrt(122))/sqrt(122), 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) = ((-1)^n)*S(2*n, 22*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-486*x+x^2).

a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=485. - Philippe Deléham, Nov 18 2008

Sum_{n>=0} 1/(a(n)+1) = sqrt(122)/22. - Amiram Eldar, Jan 01 2026

Example

(x, y) = (11*1 = 11, 1), (5357 = 11*487, 485), (2603491 = 11*236681, 235709), ... give the positive integer solutions to x^2 - 122*y^2 = -1.

Mathematica

LinearRecurrence[{486, -1}, {1, 485}, 20] (* Ray Chandler, Aug 12 2015 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-486*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 485]; [n le 2 select I[n] else 486*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(SageMath) ((1-x)/(1-486*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 485];; for n in [3..20] do a[n]:=486*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Cf. A097765 for S(n, 486).

Row 11 of array A188647.

Cf. A049310, A053120, A097766.

Sequence in context: A031722 A156774 A031632 * A031520 A235525 A249227

Adjacent sequences: A097764 A097765 A097766 * A097768 A097769 A097770

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved