A099036 a(n) = 2^n - Fibonacci(n).

1, 1, 3, 6, 13, 27, 56, 115, 235, 478, 969, 1959, 3952, 7959, 16007, 32158, 64549, 129475, 259560, 520107, 1041811, 2086206, 4176593, 8359951, 16730848, 33479407, 66987471, 134021310, 268117645, 536356683, 1072909784, 2146137379, 4292788987, 8586410014

Offset: 0

Author

Paul Barry, Sep 23 2004

Keywords

• easy
• nonn

Comments

Binomial transform of (-1)^n*A000045(n) + 1 = (-1)^n*A008346(n).

Number of compositions of n+1 that contain 1 as a part. - Vladeta Jovovic, Sep 26 2004

Generated from iterates of M * [1,1,1,...], where M = a tridiagonal matrix with [0,1,1,1,...] as the main diagonal, [1,1,1,...] as the superdiagonal and [1,0,0,0,...] as the subdiagonal. - Gary W. Adamson, Jan 05 2009

Starting with offset 1, generated from iterates of M * [1,1,1,...], M*ANS -> M*ANS,...; where M = = a tridiagonal matrix with (0,1,1,1,...) in the main diagonal, (1,1,1,...) in the superdiagonal and (1,0,0,0,...) in the subdiagonal. - Gary W. Adamson, Jan 04 2009

An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A027934 (without the leading 0). - Johannes W. Meijer, Aug 15 2010

Number of fixed points in all compositions of n+1. - Alois P. Heinz, Jun 18 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Crossrefs

Cf. A000045, A000079, A003056, A008346, A027934, A101220, A117591, A175655, A212323, A238350.

Programs

Haskell

a099036 n = a099036_list !! n
a099036_list = zipWith (-) a000079_list a000045_list
-- Reinhard Zumkeller, Aug 15 2013

Magma

[2^n-Fibonacci(n): n in [0..35]]; // Vincenzo Librandi, May 03 2011

Mathematica

Table[2^n-Fibonacci[n],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

PARI

a(n)=2^n-fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

SageMath

def A099036(n): return 2**n -fibonacci(n) # G. C. Greubel, Jun 05 2025

Formula

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

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).

a(n) = A101220(1,2,n+1) - A101220(1,2,n). - Ross La Haye, Aug 05 2005

a(n) = A000079(n+1) - A117591(n) = A117591(n) - 2 * A000045(n). - Reinhard Zumkeller, Aug 15 2013

a(n) = Sum_{t_1+2*t_2+...+n*t_n = n} multinomial(1+t_1+t_2+...+t_n, 1+t_1, t_2, ..., t_n). - Mircea Merca, Oct 09 2013

a(n) = Sum_{k=0..A003056(n+1)} k * A238350(n+1,k). - Alois P. Heinz, Jun 18 2020

E.g.f.: cosh(2*x) + sinh(2*x) - 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jan 31 2023

Extensions

More terms from Ross La Haye, Aug 05 2005