A154247 a(n) = ( (6 + sqrt(7))^n - (6 - sqrt(7))^n )/(2*sqrt(7)).

1, 12, 115, 1032, 9049, 78660, 681499, 5896848, 50998705, 440975868, 3812747971, 32964675480, 285006414601, 2464101386292, 21304030612075, 184189427142432, 1592456237959009, 13767981468377580, 119034546719719699

Offset

1,2

Comments

Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(7) = 8.6457513110....

Links

G. C. Greubel, Table of n, a(n) for n = 1..750

Index entries for linear recurrences with constant coefficients, signature (12, -29).

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 12*a(n-1) - 29*a(n-2) for n > 1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 12*x + 29*x^2). (End)

E.g.f.: sinh(sqrt(7)*x)*exp(6*x)/sqrt(7). - Ilya Gutkovskiy, Sep 08 2016

Mathematica

Join[{a=1, b=12}, Table[c=12*b-29*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
With[{c=Sqrt[7]}, Simplify/@Table[((6+c)^n-(6-c)^n)/(2c), {n, 20}]] (* or *) LinearRecurrence[{12, -29}, {1, 12}, 20] (* Harvey P. Dale, Mar 02 2012 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
(Magma) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 08 2016
(Sage) [lucas_number1(n, 12, 29) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

Crossrefs

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

Sequence in context: A090250 A199702 A307820 * A016204 A238929 A166777

Adjacent sequences: A154244 A154245 A154246 * A154248 A154249 A154250

Keyword

nonn

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