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