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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

Binomial transform of A001792.

Cf. A000244, A036290, A050914, A112626.

Programs

Magma

[ n*3^(n-4): n in [3..30] ]; // Vincenzo Librandi, Aug 19 2011

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 *)

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

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