A171475 a(n) = 6*a(n-1) - 8*a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.

1, 6, 27, 114, 468, 1896, 7632, 30624, 122688, 491136, 1965312, 7862784, 31454208, 125822976, 503304192, 2013241344, 8053014528, 32212156416, 128848822272, 515395682304, 2061583515648, 8246335635456, 32985345687552

Offset: 0

Author

Klaus Brockhaus, Dec 09 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A037480; second binomial transform of A133600.

First differences of A080960.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (6,-8).

Crossrefs

Cf. A037480 ((5*3^n +(-1)^n -6)/8), A133600 (row sums of triangle A133599), A080960 (third binomial transform of A010685).

Programs

Magma

I:=[6,27]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 02 2021

Mathematica

Table[If[n==0, 1, 3*(5*4^n - 2*2^n)/8],{n,0,30}] (* G. C. Greubel, Dec 02 2021 *)
LinearRecurrence[{6,-8},{1,6,27},30] (* Harvey P. Dale, Oct 25 2023 *)

PARI

{m=21; v=concat([1, 6, 27], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]); v}

Sage

[1]+[3*(5*4^n - 2*2^n)/8 for n in (1..30)] # G. C. Greubel, Dec 02 2021

Formula

a(n) = 3*(5*4^n - 2*2^n)/8 for n > 0.

G.f.: (1-x)*(1+x)/((1-2*x)*(1-4*x)).

E.g.f.: (1/8)*(-1 - 6*exp(2*x) + 15*exp(4*x)). - G. C. Greubel, Dec 02 2021