A088320 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5.

1, 5, 51, 515, 5201, 52525, 530451, 5357035, 54100801, 546365045, 5517751251, 55723877555, 562756526801, 5683289145565, 57395647982451, 579639768970075, 5853793337683201, 59117573145802085, 597029524795704051, 6029412821102842595, 60891157735824130001, 614940990179344142605

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..990

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (10,1).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5.

a(n) = ((5+sqrt(26))^n + (5-sqrt(26))^n)/2.

a(n) = A086927(n)/2.

Lim_{n -> oo} a(n+1)/a(n) = (5+sqrt(26)) = 10.099019... .

Lim_{n -> oo} a(n)/a(n+1) = 1/(5+sqrt(26)) = (sqrt(26)-5) = 0.099019... .

From Paul Barry, Nov 15 2003: (Start)

E.g.f.: exp(5*x)*cosh(sqrt(26)*x).

a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*26^k*5^(n-2*k).

a(n) = (-i)^n * T(n, 5*i), with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)

G.f.: (1-5*x)/(1-10*x-x^2). - R. J. Mathar, Sep 11 2008

Mathematica

LinearRecurrence[{10, 1}, {1, 5}, 31] (* Harvey P. Dale, Dec 25 2021 *)

Prog

(Magma) [n le 2 select 5^(n-1) else 10*Self(n-1) + Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 12 2022
(SageMath)
A088320=BinaryRecurrenceSequence(10, 1, 1, 5)
[A088320(n) for n in range(31)] # G. C. Greubel, Dec 12 2022

Crossrefs

Cf. A041040, A041043, A064019, A077392, A086927.

Sequence in context: A106415 A212819 A041040 * A223002 A370172 A180511

Adjacent sequences: A088317 A088318 A088319 * A088321 A088322 A088323

Keyword

easy,nonn

Author

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

Status

approved