# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a006234 Showing 1-1 of 1 %I A006234 M3496 #100 Mar 28 2024 23:53:53 %S A006234 1,4,15,54,189,648,2187,7290,24057,78732,255879,826686,2657205, %T A006234 8503056,27103491,86093442,272629233,860934420,2711943423,8523250758, %U A006234 26732013741,83682825624,261508830075,815907549834,2541865828329 %N A006234 a(n) = n*3^(n-4). %C A006234 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 A006234 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 A006234 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 A006234 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 A006234 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 A006234 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 A006234 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 A006234 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A006234 Vincenzo Librandi, Table of n, a(n) for n = 3..1000 %H A006234 Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. %H A006234 Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020. %H A006234 Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. %H A006234 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. %H A006234 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 %H A006234 Eric Weisstein's World of Mathematics, Book Graph. %H A006234 Eric Weisstein's World of Mathematics, Spanning Tree. %H A006234 Index entries for linear recurrences with constant coefficients, signature (6,-9). %F A006234 G.f.: (1-2*x)/(1-3*x)^2. - _Simon Plouffe_ in his 1992 dissertation. %F A006234 a(n+3) = Sum_{k=0..n} A112626(n, k). - _Ross La Haye_, Jan 11 2006 %F A006234 G.f.: Hypergeometric2F1([1,4],[3],3*x). - _R. J. Mathar_, Aug 09 2015 %F A006234 From _Amiram Eldar_, Jan 18 2021: (Start) %F A006234 Sum_{n>=1} 1/a(n) = 81*log(3/2). %F A006234 Sum_{n>=1} (-1)^(n+1)/a(n) = 81*log(4/3). (End) %F A006234 E.g.f.: x*(exp(3*x) - 3*x - 1)/27. - _Stefano Spezia_, Mar 04 2023 %F A006234 E.g.f. (with offset 0): exp(3*x)*(1+x). - _Enrique Navarrete_, Mar 14 2024 %e A006234 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 A006234 Table[n 3^(n-4), {n, 3, 30}] (* or *) %t A006234 CoefficientList[Series[(1-2 x)/(1-3 x)^2, {x,0,30}], x] (* _Michael De Vlieger_, May 28 2017 *) %t A006234 LinearRecurrence[{6,-9},{1,4},30] (* _Harvey P. Dale_, Aug 17 2020 *) %o A006234 (Magma) [ n*3^(n-4): n in [3..30] ]; // _Vincenzo Librandi_, Aug 19 2011 %o A006234 (PARI) a(n)=n*3^(n-4) \\ _Charles R Greathouse IV_, Sep 24 2015 %o A006234 (SageMath) [n*3^(n-4) for n in range(3,31)] # _G. C. Greubel_, Dec 27 2023 %Y A006234 Binomial transform of A001792. %Y A006234 Cf. A000244, A036290, A050914, A112626. %K A006234 nonn,easy %O A006234 3,2 %A A006234 _N. J. A. Sloane_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE