A146966 a(n) = ((6 + sqrt(7))^n + (6 - sqrt(7))^n) / 2.

1, 6, 43, 342, 2857, 24366, 209539, 1807854, 15617617, 134983638, 1166892763, 10088187654, 87218361721, 754062898686, 6519422294323, 56365243469982, 487319675104417, 4213244040623526, 36426657909454219, 314935817735368374

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = 12*a(n-1)-29*a(n-2), a(0)=1, a(1)=6. - Philippe Deléham, Nov 05 2008

G.f.: (1-6*x)/(1-12*x+29*x^2). - Klaus Brockhaus, Nov 05 2008

a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*7^(n-k))/6^n. - Philippe Deléham, Nov 06 2008

E.g.f.: exp(6*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Example

a(3) = ((6 + sqrt(7))^3 + (6 - sqrt(7))^3) / 2 = 342.

Maple

f:= gfun:-rectoproc({a(n) = 12*a(n-1)-29*a(n-2), a(0)=1, a(1)=6}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Feb 01 2016

Mathematica

RecurrenceTable[{a[1]==1, a[2]==6, a[n]== 12 a[n-1] - 29 a[n-2]}, a, {n, 20}] (* Vincenzo Librandi, Jan 31 2016 *)
LinearRecurrence[{12, -29}, {1, 6}, 20] (* Harvey P. Dale, Apr 17 2018 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r7>:=NumberField(x^2-7); S:=[ ((6+r7)^n+(6-r7)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008
(Magma) I:=[1, 6]; [n le 2 select I[n] else 12*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 31 2016
(PARI) Vec((1-6*x)/(1-12*x+29*x^2) + O(x^30)); \\ Michel Marcus, Jan 31 2016
(Sage)
def A146966_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-6*x)/(1-12*x+29*x^2) ).list()
A146966_list(20) # G. C. Greubel, Jan 08 2020
(GAP) a:=[1, 6];; for n in [3..20] do a[n]:12*a[n-1]-29*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Crossrefs

Sequence in context: A153397 A005786 A071541 * A240653 A220097 A090010

Adjacent sequences: A146963 A146964 A146965 * A146967 A146968 A146969

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

Extensions

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008

Typo in name corrected by Sean Reeves, Dec 19 2015

Status

approved