A086972 -id:A086972 - cp's OEIS frontend

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

Paul Barry, Mar 03 2003

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

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).

Cf. A001787, A001792, A081039, A081040, A086972, A200139.
First differences of A027471.
Cf. A269914, A269915, A269916, A269917.

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

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

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

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)
E.g.f.: exp(3*x)*(1 + 2*x). - Stefano Spezia, Jan 31 2025

A199923 a(n) = Sum_{k=0..3^(n-1)} gcd(k,3^(n-1)) for n > 0 and a(0) = 1.

Original entry on oeis.org

1, 2, 8, 30, 108, 378, 1296, 4374, 14580, 48114, 157464, 511758, 1653372, 5314410, 17006112, 54206982, 172186884, 545258466, 1721868840, 5423886846, 17046501516, 53464027482, 167365651248, 523017660150, 1631815099668, 5083731656658, 15816054042936

Offset: 0

Peter Luschny, Nov 12 2011

			a(3) = 9 + 1 + 1 + 3 + 1 + 1 + 3 + 1 + 1 + 9.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9).

First differences of A086972.

Row sums of A199922.

[1] cat [n le 2 select 2^(2*n-1) else 6*Self(n-1) -9*Self(n-2): n in [1..40]]; // G. C. Greubel, Nov 24 2023

LinearRecurrence[{6,-9}, {1,2,8}, 41] (* G. C. Greubel, Nov 24 2023 *)

[(2*(n+2)*3^n + 5*int(n==0))//9 for n in range(41)] # G. C. Greubel, Nov 24 2023

a(n) = 2 * (n+2) * 3^(n-2), n > 0. - Sean A. Irvine, Jun 27 2022
From G. C. Greubel, Nov 24 2023: (Start)
a(n) = (2*(n+2)*3^n + 5*[n=0])/9.
G.f.: (1-4*x+5*x^2)/(1-3*x)^2. [corrected by Jason Yuen, Oct 24 2024]
E.g.f.: (1/9)*( 5 + 2*(2 + 3*x)*exp(3*x) ). (End)

More terms from Sean A. Irvine, Jun 27 2022

cp's OEIS Frontend

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

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