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

Author

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

Keywords

• nonn

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).

Crossrefs

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

Programs

Magma

Z:=PolynomialRing(Integers()); N:=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

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 *)

Sage

[lucas_number1(n,12,29) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

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

Extensions

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

Edited by Klaus Brockhaus, Oct 06 2009