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%I A081038 #48 May 05 2023 12:18:40
%S A081038 1,5,21,81,297,1053,3645,12393,41553,137781,452709,1476225,4782969,
%T A081038 15411789,49424013,157837977,502211745,1592728677,5036466357,
%U A081038 15884240049,49977243081,156905298045,491636600541,1537671920841
%N A081038 3rd binomial transform of (1,2,0,0,0,0,0,0,...).
%C A081038 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
%C A081038 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
%C A081038 Binomial transform of {A001787(n + 1)}_{n >= 0}. - _Wolfdieter Lang_, Oct 01 2019
%H A081038 Vincenzo Librandi, <a href="/A081038/b081038.txt">Table of n, a(n) for n = 0..400</a>
%H A081038 Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.
%H A081038 Silvana Ramaj, <a href="https://digitalcommons.georgiasouthern.edu/etd/2273">New Results on Cyclic Compositions and Multicompositions</a>, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.
%H A081038 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).
%F A081038 G.f.: (1-x)/(1-3*x)^2.
%F A081038 a(n) = 6*a(n-1) - 9*a(n-2), with a(0)=1, a(1)=5.
%F A081038 a(n) = (2*n+3)*3^(n-1).
%F A081038 a(n) = Sum_{k=0..n} (k+1)*2^k*binomial(n, k).
%F A081038 a(n) = 2*A086972(n) - 1. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 10 2008
%F A081038 From _Amiram Eldar_, May 17 2022: (Start)
%F A081038 Sum_{n>=0} 1/a(n) = 9*(sqrt(3)*arctanh(1/sqrt(3)) - 1).
%F A081038 Sum_{n>=0} (-1)^n/a(n) = 9 - 3*sqrt(3)*Pi/2. (End)
%p A081038 A081038:=n->(2*n+3)*3^(n-1): seq(A081038(n), n=0..30); # _Wesley Ivan Hurt_, Mar 07 2016
%t A081038 LinearRecurrence[{6,-9},{1,5},40] (* _Harvey P. Dale_, Jun 22 2012 *)
%o A081038 (Magma) [(2*n+3)*3^(n-1): n in [0..30]]; // _Vincenzo Librandi_, Jun 09 2011
%Y A081038 Cf. A001787, A001792, A081039, A081040, A086972.
%Y A081038 First differences of A027471.
%Y A081038 Cf. A269914, A269915, A269916, A269917.
%K A081038 easy,nonn
%O A081038 0,2
%A A081038 _Paul Barry_, Mar 03 2003

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