A146964 a(n) = ((4 + sqrt(7))^n + (4 - sqrt(7))^n)/2.

1, 4, 23, 148, 977, 6484, 43079, 286276, 1902497, 12643492, 84025463, 558412276, 3711069041, 24662841844, 163903113383, 1089259330468, 7238946623297, 48108239012164, 319715392487639, 2124748988791636, 14120553377944337

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A146963.

Inverse binomial transform of A146965.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (8,-9).

Crossrefs

Cf. A098158, A146963, A146965.

Programs

GAP

a:=[1,4];; for n in [3..25] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((4+r7)^n+(4-r7)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

Maple

seq(coeff(series((1-4*x)/(1-8*x+9*x^2), x, n+1), x, n), n = 0..25); # G. C. Greubel, Jan 08 2020

Mathematica

LinearRecurrence[{8,-9}, {1,4}, 25] (* G. C. Greubel, Jan 08 2020 *)

PARI

my(x='x+O('x^25)); Vec((1-4*x)/(1-8*x+9*x^2)) \\ G. C. Greubel, Jan 08 2020

Sage

def A146964_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-4*x)/(1-8*x+9*x^2) ).list()
A146964_list(25) # G. C. Greubel, Jan 08 2020

Formula

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

a(n) = 8*a(n-1) - 9*a(n-2) with a(0)=1, a(1)=4.

G.f.: (1-4*x)/(1-8*x+9*x^2). (End)

a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*7^(n-k))/4^n. - Philippe Deléham, Nov 06 2008

E.g.f.: exp(4*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Extensions

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

Edited by Klaus Brockhaus, Jul 16 2009