A122117 a(n) = 3*a(n-1) + 4*a(n-2), with a(0)=1, a(1)=2.

1, 2, 10, 38, 154, 614, 2458, 9830, 39322, 157286, 629146, 2516582, 10066330, 40265318, 161061274, 644245094, 2576980378, 10307921510, 41231686042, 164926744166, 659706976666, 2638827906662, 10555311626650, 42221246506598

Offset: 0

Author

Philippe Deléham, Oct 19 2006

Keywords

• nonn
• easy

Comments

Inverse binomial transform of A005053. Binomial transform of [1, 1, 7, 13, 55, ...] = A015441(n+1).

Convolved with [1, 2, 2, 2, ...] = powers of 4: [1, 4, 16, 64, ...]. - Gary W. Adamson, Jun 02 2009

a(n) is the number of compositions of n when there are 2 types of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,4).

Crossrefs

Cf. A055380, A100088, A108981, A122016, A201455.

Programs

GAP

a:=[1,2];; for n in [3..30] do a[n]:=3*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, May 18 2019

Magma

I:=[1, 2]; [n le 2 select I[n] else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 06 2012

Mathematica

CoefficientList[Series[(1-x)/(1-3*x-4*x^2),{x,0,30}],x] (* Vincenzo Librandi, Jul 06 2012 *)

PARI

Vec((1-x)/(1-3*x-4*x^2)+O(x^30)) \\ Charles R Greathouse IV, Jan 11 2012

Python

def A122117(n): return ((4<<(m:=n<<1))|2)//5-((1<Chai Wah Wu, Apr 22 2025

Sage

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,3,4, lambda n: 0); [next(it) for i in range(24)] # Zerinvary Lajos, Jul 03 2008

Sage

((1-x)/(1-3*x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 18 2019

Formula

a(n) = 2*A108981(n-1) for n > 0, with a(0) = 1.

a(2*n) = 4*a(2*n-1) + 2, a(2*n+1) = 4*a(2*n) - 2.

a(n) = Sum_{k=0..n} 2^(n-k)*A055380(n,k).

G.f.: (1-x)/(1-3*x-4*x^2).

Lim_{n->infinity} a(n+1)/a(n) = 4.

a(n) = Sum_{k=0..n} A122016(n,k)*2^k. - Philippe Deléham, Nov 05 2008

a(n) = A100088(2*n). - Chai Wah Wu, Apr 22 2025

Extensions

Corrected by T. D. Noe, Nov 07 2006