A006234 a(n) = n*3^(n-4). (Formerly M3496)

1, 4, 15, 54, 189, 648, 2187, 7290, 24057, 78732, 255879, 826686, 2657205, 8503056, 27103491, 86093442, 272629233, 860934420, 2711943423, 8523250758, 26732013741, 83682825624, 261508830075, 815907549834, 2541865828329

Offset

3,2

Comments

For n >= 1 a(n) is also the determinant of the n-3 X n-3 matrix with 4's on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

a(n+3) = det(M(n)) where M(n) is the n X n matrix with m(i,i) = 4, m(i,j) = i/j for i != j. - Benoit Cloitre, Feb 01 2003

Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 2*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

a(n+3) is the number of words of length n on {A, B, C, D} with no D appearing anywhere to the right of an A. - Rob Pratt, Aug 04 2004

Number of spanning trees in the book graph of order n-2, i.e., S_{n-2} X P_2 (S_k = the star graph on k nodes) (conjectured). This conjecture is true - see Doslic (2013). - N. J. A. Sloane, Dec 28 2013

Conjecture: a(n+2) is the total number of parts used in the compositions of n if the parts can be runs of any length from 1 to n, and contain any integers from 1 to n. (The number of such compositions is given by A000244(n-1).) - Gregory L. Simay, May 27 2017

a(n+3) is the number of words of length n defined on 4 letters where one of the letters is used at most once. - Enrique Navarrete, Mar 14 2024

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..1000

Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.

Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.

Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Eric Weisstein's World of Mathematics, Book Graph.

Eric Weisstein's World of Mathematics, Spanning Tree.

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

Formula

G.f.: (1-2*x)/(1-3*x)^2. - Simon Plouffe in his 1992 dissertation.

a(n+3) = Sum_{k=0..n} A112626(n, k). - Ross La Haye, Jan 11 2006

G.f.: Hypergeometric2F1([1,4],[3],3*x). - R. J. Mathar, Aug 09 2015

From Amiram Eldar, Jan 18 2021: (Start)

Sum_{n>=1} 1/a(n) = 81*log(3/2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 81*log(4/3). (End)

E.g.f.: x*(exp(3*x) - 3*x - 1)/27. - Stefano Spezia, Mar 04 2023

E.g.f. (with offset 0): exp(3*x)*(1+x). - Enrique Navarrete, Mar 14 2024

Example

For n=3, the total number of parts is (3+2)3^(3+2-4)=(5)(3)=15 (each part indicated by "[]"): [3]; [2,1]; [1,2]; [2],[1]; [1],[2]; [1,1,1]; [1,1],[1]; [1],[1,1]; [1],[1],[1]. Note that these 15 parts are arranged into 9 = A000244(3-1)compositions. - Gregory L. Simay, May 27 2017

Mathematica

Table[n 3^(n-4), {n, 3, 30}] (* or *)
CoefficientList[Series[(1-2 x)/(1-3 x)^2, {x, 0, 30}], x] (* Michael De Vlieger, May 28 2017 *)
LinearRecurrence[{6, -9}, {1, 4}, 30] (* Harvey P. Dale, Aug 17 2020 *)

Prog

(Magma) [ n*3^(n-4): n in [3..30] ]; // Vincenzo Librandi, Aug 19 2011
(PARI) a(n)=n*3^(n-4) \\ Charles R Greathouse IV, Sep 24 2015
(SageMath) [n*3^(n-4) for n in range(3, 31)] # G. C. Greubel, Dec 27 2023

Crossrefs

Binomial transform of A001792.

Cf. A000244, A036290, A050914, A112626.

Sequence in context: A227382 A090326 A291032 * A094821 A071723 A001559

Adjacent sequences: A006231 A006232 A006233 * A006235 A006236 A006237

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved