A154245 a(n) = ( (4 + sqrt(7))^n - (4 - sqrt(7))^n )/(2*sqrt(7)).

1, 8, 55, 368, 2449, 16280, 108199, 719072, 4778785, 31758632, 211059991, 1402652240, 9321678001, 61949553848, 411701328775, 2736064645568, 18183205205569, 120841059834440, 803079631825399, 5337067516093232

Offset

1,2

Comments

Second binomial transform of A109115.

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(7) = 6.6457513110....

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Index entries for linear recurrences with constant coefficients, signature (8,-9).

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 8*a(n-1) - 9*a(n-2) for n > 1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 8*x + 9*x^2). (End)

a(n) = b such that (3^(n-1)/2)*Integral_{x=0..Pi/2} (sin(n*x))/(4/3-cos(x)) dx = c + b*log(2). - Francesco Daddi, Aug 02 2011

E.g.f.: (1/sqrt(7))*exp(4*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 07 2016

Mathematica

Table[Simplify[((4+Sqrt[7])^n -(4-Sqrt[7])^n)/(2*Sqrt[7])], {n, 30}] (* or *) LinearRecurrence[{8, -9}, {1, 8}, 30] (* G. C. Greubel, Sep 07 2016 *)
Rest@ CoefficientList[Series[x/(1 -8x +9x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 08 2016 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
(Magma) I:=[1, 8]; [n le 2 select I[n] else 8*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 08 2016
(Sage) [lucas_number1(n, 8, 9) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009
(PARI) my(x='x+O('x^30)); Vec( x/(1-8*x+9*x^2) ) \\ G. C. Greubel, May 21 2019
(GAP) a:=[1, 8];; for n in [3..30] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Crossrefs

Equals (A094432 without initial term 0)/3.

Cf. A010465 (decimal expansion of square root of 7), A109115.

Sequence in context: A026994 A110184 A013698 * A143420 A075734 A033890

Adjacent sequences: A154242 A154243 A154244 * A154246 A154247 A154248

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

Status

approved