A140960 a(n) = (2*(-1)^n - 2^(n+1) + 3*n*2^n)/9.

0, 0, 2, 6, 18, 46, 114, 270, 626, 1422, 3186, 7054, 15474, 33678, 72818, 156558, 334962, 713614, 1514610, 3203982, 6757490, 14214030, 29826162, 62448526, 130489458, 272163726, 566697074, 1178133390, 2445745266, 5070447502, 10498808946, 21713445774, 44858547314

Offset: 0

Author

Paul Curtz, Jul 26 2008

Keywords

• nonn
• easy

Comments

Specify that a triangle has T(n,0) = T(n,n) = A001045(n), and T(r,c) = T(r-1,c-1) + T(r-1,c). The sum of the terms in the first n rows is a(n+1). - J. M. Bergot, May 21 2013

a(n) is the difference between the total number of runs of equal parts in the compositions of n+1, and the compositions of n+1. - Gregory L. Simay, May 04 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)

Programs

Magma

[( 2*(-1)^n-2^(n+1)+3*n*2^n)/9: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

Mathematica

LinearRecurrence[{3,0,-4},{0,0,2},40] (* Harvey P. Dale, Apr 14 2015 *)

PARI

a(n)=(2*(-1)^n-2^(n+1)+3*n*2^n)/9 \\ Charles R Greathouse IV, Oct 16 2015

Formula

a(n+1) - 2*a(n) = A078008(n+1) = 2*A001045(n).

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

a(n) = 2*A045883(n-1).

a(n) = 3*a(n-1) - 4*a(n-3), n > 2.

a(n) = A059570(n+1) - A011782(n+1). - Gregory L. Simay, May 04 2017

Extensions

Definition replaced with Lava's closed form of August 2008 by R. J. Mathar, Feb 11 2010