A088316 a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.

2, 13, 171, 2236, 29239, 382343, 4999698, 65378417, 854919119, 11179326964, 146186169651, 1911599532427, 24996980091202, 326872340718053, 4274337409425891, 55893258663254636, 730886700031736159, 9557420359075824703, 124977351368017457298, 1634262988143302769577

Offset: 0

Author

Nikolay V. Kosinov, Dmitry V. Polyakov (kosinov(AT)unitron.com.ua), Nov 06 2003

Keywords

• nonn
• easy

Comments

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..890
• Tanya Khovanova, Recursive Sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (13,1).

Crossrefs

Cf. A006905, A041318, A054413, A086902, A088316, A097845, A140455.

Programs

Magma

I:=[2,13]; [n le 2 select I[n] else 13*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

Mathematica

LinearRecurrence[{13,1}, {2,13}, 31] (* Stefano Spezia, Sep 20 2022 *)

SageMath

A088316=BinaryRecurrenceSequence(13,1,2,13)
[A088316(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

Formula

a(n) = ((13+sqrt(173))/2)^n + ((13-sqrt(173))/2)^n.

Lim_{n -> oo} a(n+1)/a(n) = (13 + sqrt(173))/2.

Lim_{n -> oo} a(n)/a(n+1) = 2/(13+sqrt(173)).

G.f.: (2-13*x)/(1-13*x-x^2). - Philippe Deléham, Nov 02 2008

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2*n+1) = 13*A097845(n).

a(3*n+1) = A041318(5n), a(3n+2) = A041318(5n+3), a(3n+3) = 2*A041318(5n+4).

Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.

Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)