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URL: https://oeis.org/A154247

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A154247
a(n) = ( (6 + sqrt(7))^n - (6 - sqrt(7))^n )/(2*sqrt(7)).
1
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....
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
(SageMath) [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
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