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

Author

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

Keywords

• nonn

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).

Crossrefs

Cf. A098158, A108851, A146964.

Programs

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

Magma

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

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 *)

PARI

my(x='x+O('x^25)); Vec((1-3*x)/(1-6*x+2*x^2)) \\ G. C. Greubel, Jan 08 2020

Sage

def A146963_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x)/(1-6*x+2*x^2) ).list()
A146963_list(25) # G. C. Greubel, Jan 08 2020

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

Extensions

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

Edited by Klaus Brockhaus, Jul 16 2009