A146963 a(n) = ((3 + sqrt(7))^n + (3 - sqrt(7))^n)/2.

1, 3, 16, 90, 508, 2868, 16192, 91416, 516112, 2913840, 16450816, 92877216, 524361664, 2960415552, 16713769984, 94361788800, 532743192832, 3007735579392, 16980927090688, 95870091385344, 541258694130688

Offset

0,2

Comments

Binomial transform of A108851.

Inverse binomial transform of A146964.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..158

Index entries for linear recurrences with constant coefficients, signature (6,-2).

Formula

From Philippe Deléham and Klaus Brockhaus, Nov 05 2008: (Start)

a(n) = 6*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=3.

G.f.: (1-3*x)/(1-6*x+2*x^2). (End)

a(n) = (Sum_{k=0..n} A098158(n,k)*3^(2*k)*7^(n-k))/3^n. - Philippe Deléham, Nov 06 2008

E.g.f.: exp(3*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

a(n) = A154244(n)-3*A154244(n-1). - R. J. Mathar, Jan 25 2023

Maple

seq(coeff(series((1-3*x)/(1-6*x+2*x^2), x, n+1), x, n), n = 0..25); # G. C. Greubel, Jan 08 2020

Mathematica

Transpose[NestList[Join[{Last[#], 6Last[#]-2First[#]}]&, {1, 3}, 25]] [[1]]  (* or *) CoefficientList[Series[(1-3x)/(1-6x+2x^2), {x, 0, 25}], x]  (* Harvey P. Dale, Apr 11 2011 *)
LinearRecurrence[{6, -2}, {1, 3}, 25] (* G. C. Greubel, Jan 08 2020 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r7>:=NumberField(x^2-7); S:=[ ((3+r7)^n+(3-r7)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008
(PARI) my(x='x+O('x^25)); Vec((1-3*x)/(1-6*x+2*x^2)) \\ G. C. Greubel, Jan 08 2020
(SageMath)
def A146963_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x)/(1-6*x+2*x^2) ).list()
A146963_list(25) # G. C. Greubel, Jan 08 2020
(GAP) a:=[1, 3];; for n in [3..25] do a[n]:=6*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Crossrefs

Cf. A098158, A108851, A146964.

Sequence in context: A347930 A026111 A026330 * A074562 A130744 A344455

Adjacent sequences: A146960 A146961 A146962 * A146964 A146965 A146966

Keyword

nonn

Author

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

Extensions

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

Edited by Klaus Brockhaus, Jul 16 2009

Status

approved