A108981 a(n) = 3a(n-1) + 4a(n-2), a(0) = 1, a(1) = 5.

1, 5, 19, 77, 307, 1229, 4915, 19661, 78643, 314573, 1258291, 5033165, 20132659, 80530637, 322122547, 1288490189, 5153960755, 20615843021, 82463372083, 329853488333, 1319413953331, 5277655813325, 21110623253299

Offset: 0

Author

Philippe Deléham, Jul 23 2005

Keywords

• nonn
• easy

Comments

The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Apr 15 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-2, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = charpoly(A,2). - Milan Janjic, Jan 26 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (3,4).

Crossrefs

Cf. A003948, A037481.

Programs

Magma

I:=[1,5]; [n le 2 select I[n] else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 17 2014

Mathematica

LinearRecurrence[{3,4},{1,5},30] (* Harvey P. Dale, Feb 16 2014 *)

PARI

Vec((1+2*x)/(1+x)/(1-4*x)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

Python

def A108981(n): return ((4<<(m:=n+1<<1))|2)//5-((1<>1 # Chai Wah Wu, Apr 22 2025

Formula

Inverse binomial transform of A003948.

a(2n) = 4a(2n-1) - 1; a(2n+1) = 4a(2n) + 1.

O.g.f.: (1+2*x)/((1+x)(1-4*x)). - R. J. Mathar, Apr 02 2008

Sum_{k=0..n} a(k) = A037481(n+1). - Philippe Deléham, Apr 15 2008

Extensions

Corrected by T. D. Noe, Nov 07 2006

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 14 2008