A006234 - OEIS

%I M3496 #100 Mar 28 2024 23:53:53

%S 1,4,15,54,189,648,2187,7290,24057,78732,255879,826686,2657205,

%T 8503056,27103491,86093442,272629233,860934420,2711943423,8523250758,

%U 26732013741,83682825624,261508830075,815907549834,2541865828329

%N a(n) = n*3^(n-4).

%C 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

%C 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

%C 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

%C 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

%C 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

%C 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

%C 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

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

%H Vincenzo Librandi, <a href="/A006234/b006234.txt">Table of n, a(n) for n = 3..1000</a>

%H Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.

%H Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, <a href="https://arxiv.org/abs/2003.00524">New production matrices for geometric graphs</a>, arXiv:2003.00524 [math.CO], 2020.

%H Germain Kreweras, <a href="http://dx.doi.org/10.1016/0095-8956(78)90021-7">Complexité et circuits Eulériens dans les sommes tensorielles de graphes</a>, J. Combin. Theory, B 24 (1978), 202-212.

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BookGraph.html">Book Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).

%F G.f.: (1-2*x)/(1-3*x)^2. - _Simon Plouffe_ in his 1992 dissertation.

%F a(n+3) = Sum_{k=0..n} A112626(n, k). - _Ross La Haye_, Jan 11 2006

%F G.f.: Hypergeometric2F1([1,4],[3],3*x). - _R. J. Mathar_, Aug 09 2015

%F From _Amiram Eldar_, Jan 18 2021: (Start)

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

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

%F E.g.f.: x*(exp(3*x) - 3*x - 1)/27. - _Stefano Spezia_, Mar 04 2023

%F E.g.f. (with offset 0): exp(3*x)*(1+x). - _Enrique Navarrete_, Mar 14 2024

%e 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

%t Table[n 3^(n-4), {n, 3, 30}] (* or *)

%t CoefficientList[Series[(1-2 x)/(1-3 x)^2, {x,0,30}], x] (* _Michael De Vlieger_, May 28 2017 *)

%t LinearRecurrence[{6,-9},{1,4},30] (* _Harvey P. Dale_, Aug 17 2020 *)

%o (Magma) [ n*3^(n-4): n in [3..30] ]; // _Vincenzo Librandi_, Aug 19 2011

%o (PARI) a(n)=n*3^(n-4) \\ _Charles R Greathouse IV_, Sep 24 2015

%o (SageMath) [n*3^(n-4) for n in range(3,31)] # _G. C. Greubel_, Dec 27 2023

%Y Binomial transform of A001792.

%Y Cf. A000244, A036290, A050914, A112626.

%K nonn,easy

%O 3,2

%A _N. J. A. Sloane_