A083102 a(n) = 2*a(n-1) + 10*a(n-2), with a(0) = 1, a(1) = 2.

1, 2, 14, 48, 236, 952, 4264, 18048, 78736, 337952, 1463264, 6306048, 27244736, 117549952, 507547264, 2190594048, 9456660736, 40819261952, 176205131264, 760602882048, 3283257076736, 14172542973952, 61177656715264, 264080743170048, 1139938053492736, 4920683538685952

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

Keywords

• easy
• nonn

Comments

a(n+1) = a(n) + A083101(n). A083101(n)/a(n) converges to sqrt(11).

Antidiagonals of A038207. - Mark Dols, Aug 31 2009

Numerators of stationary probabilities for M2/M/1 queue system. In this queue, customers arrive in groups of 2. Intensity of arrival = 2. Service rate = 5. There is only one server and an infinite queue. - Igor Kleiner, Nov 02 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (2,10).

Crossrefs

Cf. A038207, A083101.

Programs

Magma

I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1) + 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

Mathematica

CoefficientList[Series[1/(1-2x-10x^2), {x, 0, 25}], x]
LinearRecurrence[{2,10}, {1,2}, 30] (* G. C. Greubel, Jan 08 2018 *)

PARI

x='x+O('x^30); Vec(1/(1-2*x-10*x^2)) \\ G. C. Greubel, Jan 08 2018

Sage

[lucas_number1(n,2,-10) for n in range(1, 24)] # Zerinvary Lajos, Apr 22 2009

Formula

G.f.: 1/(1-2*x-10*x^2).

From Paul Barry, Sep 29 2004: (Start)

E.g.f.: exp(x) * sinh(sqrt(11)*x) / sqrt(11).

a(n) = Sum_{k=0..n} binomial(n,2*k+1) * 11^k. (End)

a(n) = ((1+sqrt(11))^n - (1-sqrt(11))^n)/(2*sqrt(11)). - Rolf Pleisch, Jul 06 2009

G.f.: G(0)/(2-2*x), where G(k)= 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 10*x )/( x*(4*k+4 + 10*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013