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A054878 Number of closed walks of length n along the edges of a tetrahedron based at a vertex.

%I A054878 #75 May 31 2024 14:41:46
%S A054878 1,0,3,6,21,60,183,546,1641,4920,14763,44286,132861,398580,1195743,
%T A054878 3587226,10761681,32285040,96855123,290565366,871696101,2615088300,
%U A054878 7845264903,23535794706,70607384121,211822152360,635466457083
%N A054878 Number of closed walks of length n along the edges of a tetrahedron based at a vertex.
%C A054878 Number of closed walks of length n at a vertex of C_4, the cyclic graph on 4 nodes. 3*A015518(n) + a(n) = 3^n. - _Paul Barry_, Feb 03 2004
%C A054878 Form the digraph with matrix A = [0,1,1,1; 1,0,1,1; 1,1,0,1; 1,0,1,1]; a(n) corresponds to the (1,1) term of A^n. - _Paul Barry_, Oct 02 2004
%C A054878 Absolute values of A084567 (compare generating functions).
%C A054878 For n > 1, 4*a(n)=A218034(n)= the trace of the n-th power of the adjacency matrix for a complete 4-graph, a 4 X 4 matrix with a null diagonal and all ones for off-diagonal elements. The diagonal elements for the n-th power are a(n) and the off-diagonal are a(n)+1 for an odd power and a(n)-1 for an even (cf. A001045). - _Tom Copeland_, Nov 06 2012
%H A054878 Vincenzo Librandi, <a href="/A054878/b054878.txt">Table of n, a(n) for n = 0..1000</a>
%H A054878 Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL21/Choi/choi10.html">A Generalization of Collatz Functions and Jacobsthal Numbers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
%H A054878 M. Dukes and C. D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%H A054878 R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, (Nov 2020), Section 2.
%H A054878 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,3).
%F A054878 a(n) = (3^n + (-1)^n*3)/4.
%F A054878 G.f.: 1/4*(3/(1+x) + 1/(1-3*x)).
%F A054878 E.g.f.: (exp(3*x) + 3*exp(-x))/4. - _Paul Barry_, Apr 20 2003
%F A054878 a(n) = 3^n - a(n-1) with a(0)=0. - _Labos Elemer_, Apr 26 2003
%F A054878 G.f.: (1 - 3*x^2 - 2*x^3)/(1 - 6*x^2 - 8*x^3 - 3*x^4) = (1 - 3*x^2 - 2*x^3)/charpoly(adj(C_4)). - _Paul Barry_, Feb 03 2004
%F A054878 From _Paul Barry_, Oct 02 2004: (Start)
%F A054878 G.f.: (1-2*x)/(1 - 2*x - 3*x^2).
%F A054878 a(n) = 2*a(n-1) + 3*a(n-2). (End)
%F A054878 G.f.: 1 - x + x/Q(0), where Q(k) = 1 + 3*x^2 - (3*k+4)*x + x*(3*k+1 - 3*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013
%F A054878 a(n+m) = a(n)*a(m) + a(n+1)*a(m+1)/3. - _Yuchun Ji_, Sep 12 2017
%F A054878 a(n) = 3*a(n-1) + 3*(-1)^n. - _Yuchun Ji_, Sep 13 2017
%F A054878 From _Peter Bala_, May 28 2024: (Start)
%F A054878 a(n) = (-1)^n + Sum_{k = 1..n} (-1)^(n-k)*binomial(n, k)*4^(k-1).
%F A054878 G.f.: A(x) = 1/(1 - x^2) o 1/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A015575.
%F A054878 The black diamond product A(x) o A(x) is the g.f. for the number of closed walks of length n at a vertex along the edges of the 15-simplex. (End)
%p A054878 A054878:=n->(3^n + (-1)^n*3)/4: seq(A054878(n), n=0..50); # _Wesley Ivan Hurt_, Sep 16 2017
%t A054878 Table[(3^n + (-1)^n*3)/4, {n, 0, 26}] (* or *)
%t A054878 CoefficientList[Series[1/4*(3/(1 + x) + 1/(1 - 3 x)), {x, 0, 26}], x] (* _Michael De Vlieger_, Sep 15 2017 *)
%o A054878 (Magma) [(3^n+(-1)^n*3)/4: n in [0..35]]; // _Vincenzo Librandi_, Jun 30 2011
%o A054878 (PARI) a(n) = (3^n + 3*(-1)^n)/4; \\ _Altug Alkan_, Sep 17 2017
%Y A054878 Row n=4 of A109502. A084567 (signed version).
%Y A054878 {a(n)/3} for n>0 is A015518, non-closed walks.
%Y A054878 Cf. A001045, A078008, A097073, A115341, A015518 (sequences where a(n)=3^n-a(n-1)). - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%K A054878 nonn,walk,easy
%O A054878 0,3
%A A054878 Paolo Dominici (pl.dm(AT)libero.it), May 23 2000