A172481 a(n) = (3*n*2^n+2^(n+4)+2*(-1)^n)/18.

1, 2, 5, 11, 25, 55, 121, 263, 569, 1223, 2617, 5575, 11833, 25031, 52793, 111047, 233017, 487879, 1019449, 2126279, 4427321, 9204167, 19107385, 39612871, 82021945, 169636295, 350457401, 723284423, 1491308089, 3072094663, 6323146297, 13004206535, 26724240953

Offset: 0

Author

Paul Curtz, Feb 04 2010

Keywords

• nonn
• easy

Comments

The binomial transform is in A126184.

An elephant sequence, see A175654 and A175655. There are 24 A[5] vectors, with decimal values between 7 and 448, that lead for the corner squares to this sequence. Its companion sequence for the central square is A175656. Furthermore there are 36 A[5] vectors, with decimal values between 15 and 480, that lead for the central square to four times this sequence for n >= -1. Its companion sequence for the corner squares is A059570. - Johannes W. Meijer, Aug 15 2010

a(n) is also the number of runs of weakly increasing parts in all compositions of n+1. a(2) = 5: (111), (12), (2)(1), (3). - Alois P. Heinz, Apr 30 2017

Links

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

Crossrefs

Cf. A059570, A151529, A127984.

Programs

Magma

[(3*n*2^n+2^(n+4)+2*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011

Mathematica

Table[(3n 2^n+2^(n+4)+2(-1)^n)/18,{n,0,40}]  (* or *)
CoefficientList[Series[(1-x-x^2)/((1+x)(1-2x)^2), {x,0,40}], x]  (* Harvey P. Dale, Mar 28 2011 *)

PARI

a(n)=(3*n*2^n+2^(n+4)+2*(-1)^n)/18 \\ Charles R Greathouse IV, Oct 07 2015

Formula

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

a(n) = A001045(n-1)+2*a(n-1), n>0.

a(n)+A139790(n) = 2^(n+1) = A000079(n+1).

a(n) = A139790(n)+A140960(n).

a(n) = A001045(n)+(-1)^n*A084219(n).

a(n) = A127984(n) + 2^(n-1). Application: Problem 11623, AMM 119 (2012) 161. - Stephen J. Herschkorn, Feb 11 2012

Extensions

Definition replaced by explicit formula by R. J. Mathar, Feb 11 2010