A081038 3rd binomial transform of (1,2,0,0,0,0,0,0,...).

1, 5, 21, 81, 297, 1053, 3645, 12393, 41553, 137781, 452709, 1476225, 4782969, 15411789, 49424013, 157837977, 502211745, 1592728677, 5036466357, 15884240049, 49977243081, 156905298045, 491636600541, 1537671920841

Offset

0,2

Comments

a(n) is the number of distinguished parts in all compositions of n+1 in which some (possibly all or none) of the parts have been distinguished. a(1) = 2 because we have: 2', 1'+1, 1+1', 1'+1' where we see 5's marking the distinguished parts. With offset=1, a(n) = Sum_{k=1..n} A200139(n,k)*k. - Geoffrey Critzer, Jan 12 2013

For n>=1, a(n-1) the number of ternary strings of length 2n containing the block 11..12 with n ones where no runs of length larger than n are permitted. - Marko Riedel, Mar 08 2016

Binomial transform of {A001787(n + 1)}_{n >= 0}. - Wolfdieter Lang, Oct 01 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.

Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.

Index entries for linear recurrences with constant coefficients, signature (6,-9).

Formula

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

a(n) = 6*a(n-1) - 9*a(n-2), with a(0)=1, a(1)=5.

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

a(n) = Sum_{k=0..n} (k+1)*2^k*binomial(n, k).

a(n) = 2*A086972(n) - 1. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 10 2008

From Amiram Eldar, May 17 2022: (Start)

Sum_{n>=0} 1/a(n) = 9*(sqrt(3)*arctanh(1/sqrt(3)) - 1).

Sum_{n>=0} (-1)^n/a(n) = 9 - 3*sqrt(3)*Pi/2. (End)

Maple

A081038:=n->(2*n+3)*3^(n-1): seq(A081038(n), n=0..30); # Wesley Ivan Hurt, Mar 07 2016

Mathematica

LinearRecurrence[{6, -9}, {1, 5}, 40] (* Harvey P. Dale, Jun 22 2012 *)

Prog

(Magma) [(2*n+3)*3^(n-1): n in [0..30]]; // Vincenzo Librandi, Jun 09 2011

Crossrefs

Cf. A001787, A001792, A081039, A081040, A086972.

First differences of A027471.

Cf. A269914, A269915, A269916, A269917.

Sequence in context: A269916 A273205 A269917 * A273389 A153008 A292878

Adjacent sequences: A081035 A081036 A081037 * A081039 A081040 A081041

Keyword

easy,nonn

Author

Paul Barry, Mar 03 2003

Status

approved