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A054878
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Number of closed walks of length n along the edges of a tetrahedron based at a vertex.
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21
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1, 0, 3, 6, 21, 60, 183, 546, 1641, 4920, 14763, 44286, 132861, 398580, 1195743, 3587226, 10761681, 32285040, 96855123, 290565366, 871696101, 2615088300, 7845264903, 23535794706, 70607384121, 211822152360, 635466457083
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of closed walks of length n at a vertex of C_4, the cyclic graph on 4 nodes. 3*A015518(n)+A054878(n)=3^n. - Paul Barry (pbarry(AT)wit.ie), Feb 03 2004
Form the digraph with matrix A=[0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. A054878(n) corresponds to the (1,1) term of A^n. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
General form: k=3^n-k. Also: A001045, A078008, A097073, A115341, A015518 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
Absolute values of A084567 (compare generating functions).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (2,3).
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FORMULA
| a(n) = (3^n+(-1)^n*3)/4.
G.f.: 1/4*(3/(1+x)+1/(1-3*x)).
E.g.f.: (exp(3*x)+3*exp(-x))/4. - Paul Barry (pbarry(AT)wit.ie), Apr 20 2003
a(n) = 3^n - a(n-1) with a(0)=0 - Labos E. (labos(AT)ana.sote.hu), Apr 26 2003
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)); a(n) = 6*a(n-2)+8*a(n-3)+3*a(n-4). - Paul Barry (pbarry(AT)wit.ie), Feb 03 2004
G.f.: (1-2*x)/(1-2*x-3*x^2); a(n)=2*a(n-1)+3*a(n-2); a(n)=a(n-1)+5*a(n-2)+3*a(n-3). - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
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MATHEMATICA
| k=0; lst={1, k}; Do[k=3^n-k; AppendTo[lst, k], {n, 1, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
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PROG
| (MAGMA) [(3^n+(-1)^n*3)/4: n in [0..35]]; // Vincenzo Librandi, Jun 30 2011
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CROSSREFS
| {a(n)/3} for n>0 is A015518.
Cf. A001045, A078008, A097073, A115341, A015518 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
Sequence in context: A148621 A148622 A148623 * A084567 A135686 A151961
Adjacent sequences: A054875 A054876 A054877 * A054879 A054880 A054881
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KEYWORD
| nonn,walk,easy
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AUTHOR
| Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
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