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A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.

%I A000340 M3882 N1592 #90 Jan 23 2025 12:41:12
%S A000340 1,5,18,58,179,543,1636,4916,14757,44281,132854,398574,1195735,
%T A000340 3587219,10761672,32285032,96855113,290565357,871696090,2615088290,
%U A000340 7845264891,23535794695,70607384108,211822152348,635466457069
%N A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.
%C A000340 From _Johannes W. Meijer_, Feb 20 2009: (Start)
%C A000340 Second right hand column (n-m=1) of the A156920 triangle.
%C A000340 The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925.
%C A000340 (End)
%C A000340 Partial sums of A003462, and thus the second partial sums of A000244 (3^n). Also column k=2 of A106516. - _John Keith_, Jan 04 2022
%D A000340 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
%D A000340 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000340 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000340 Vincenzo Librandi, <a href="/A000340/b000340.txt">Table of n, a(n) for n = 0..1000</a>
%H A000340 Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, <a href="https://arxiv.org/abs/2501.06675">Chip-Firing on Infinite k-ary Trees</a>, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
%H A000340 Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, <a href="https://arxiv.org/abs/2411.02897">Patterns in Multi-dimensional Permutations</a>, arXiv:2411.02897 [math.CO], 2024. See p. 7.
%H A000340 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=389">Encyclopedia of Combinatorial Structures 389</a>
%H A000340 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000340 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000340 László Tóth, <a href="https://arxiv.org/abs/2002.06584">On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers</a>, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence. See also <a href="https://doi.org/10.1090/proc/14863">Proc. Amer. Math. Soc.</a> 148 (2020), 461-469.
%H A000340 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,3).
%F A000340 G.f.: 1/((1-3*x)*(1-x)^2).
%F A000340 a(n) = (3^(n+2) - 2*n - 5)/4.
%F A000340 a(n) = Sum_{k=0..n+1} (n-k+1)*3^k = Sum_{k=0..n+1} k*3^(n-k+1). - _Paul Barry_, Jul 30 2004
%F A000340 a(n) = Sum_{k=0..n} binomial(n+2, k+2)*2^k. - _Paul Barry_, Jul 30 2004
%F A000340 a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 3*a(n-2) + 1. - _Miklos Kristof_, Mar 09 2005
%F A000340 a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - _Johannes W. Meijer_, Feb 20 2009
%F A000340 a(-2 - n) = 3^-n * A014915(n). - _Michael Somos_, May 28 2014
%F A000340 E.g.f.: exp(x)*(9*exp(2*x) - 2*x - 5)/4. - _Stefano Spezia_, Nov 09 2024
%e A000340 G.f. = 1 + 5*x + 18*x^2 + 58*x^3 + 179*x^4 + 543*x^5 + 1636*x^6 + ...
%p A000340 a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n],n=0..50); # _Miklos Kristof_, Mar 09 2005
%p A000340 A000340:=-1/(3*z-1)/(z-1)**2; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A000340 a[ n_] := MatrixPower[ {{1, 0, 0}, {1, 1, 0}, {1, 1, 3}}, n + 1][[3, 1]]; (* _Michael Somos_, May 28 2014 *)
%t A000340 RecurrenceTable[{a[0]==1,a[n]==3a[n-1]+n+1},a,{n,30}] (* or *) LinearRecurrence[{5,-7,3},{1,5,18},30] (* _Harvey P. Dale_, Jan 31 2017 *)
%o A000340 (Magma) [(3^(n+2)-2*n-5)/4: n in [0..30]]; // _Vincenzo Librandi_, Aug 15 2011
%Y A000340 From _Johannes W. Meijer_, Feb 20 2009: (Start)
%Y A000340 Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924.
%Y A000340 Equals A156920 second right hand column.
%Y A000340 Equals A142963 second right hand column divided by 2^n.
%Y A000340 Equals A156919 second right hand column divided by 2.
%Y A000340 (End)
%Y A000340 Cf. A014915.
%Y A000340 Equals column k=1 of A008971 (shifted). - _Jeremy Dover_, Jul 11 2021
%Y A000340 Cf. A000340, A003462 (first differences), A106516.
%K A000340 nonn,easy
%O A000340 0,2
%A A000340 _N. J. A. Sloane_, _Simon Plouffe_