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A033191
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Binomial transform of [ 1, 0, 1, 1, 3, 6, 15, 36, 91, 231, 595,... ], which is essentially binomial(fibonacci(k) + 1, 2).
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7
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1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16778, 58598, 206516, 732825, 2613834, 9358677, 33602822, 120902914, 435668420, 1571649221, 5674201118, 20497829133, 74079051906, 267803779710, 968355724724
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 1, s(2n) = 1. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 14 2004
Contribution from Paul Barry (pbarry(AT)wit.ie), Dec 17 2009: (Start)
The sequence 1,2,5,14,... has g.f. 1/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x))))=(1-6x+10x^2-4x^3)/(1-8x+21x^2-20x^3+5x^4),
and is the second binomial transform A001519 aerated. (End)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
Counts all paths of length (2*n), n>=0, starting and ending at the initial node on the path graph P_9, see the Maple program.
(End)
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FORMULA
| G.f.: (1-7x+15x^2-10x^3+x^4)/(1-8x+21x^2-20x^3+5x^4). - Ralf Stephan (ralf(AT)ark.in-berlin.de), May 13 2003
a(n)=(1/5)*Sum(r, 1, 9, Sin(r*Pi/10)^2(2Cos(r*Pi/10))^(2n)), n>=1 a(n)=8a(n-1)-21a(n-2)+20a(n-3)-5a(n-4), n>=5 - Herbert Kociemba (kociemba(AT)t-online.de), Jun 14 2004
a(2)=1, a(3)=2, a(4)=5, a(5)=14, a(n)=8a(n-1)-21a(n-2)+20a(n-3)- 5a(n-4) [From Harvey P. Dale, Apr 26 2011]
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MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):=B(n)[1, 1]; od: seq(a(2*n), n=0..nmax);
(End)
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MATHEMATICA
| CoefficientList[Series[(1-7x+15x^2-10x^3+x^4)/(1-8x+21x^2-20x^3+5x^4), {x, 0, 30}], x] (* or *) Join[{1}, LinearRecurrence[{8, -21, 20, -5}, {1, 2, 5, 14}, 30]] (* From Harvey P. Dale, Apr 26 2011 *)
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CROSSREFS
| Cf. A033192.
Cf. A081567, A147748 and A178381.
Sequence in context: A080938 A054394 A036769 * A168491 A115140 A120588
Adjacent sequences: A033188 A033189 A033190 * A033192 A033193 A033194
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KEYWORD
| nonn
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AUTHOR
| Simon Norton (simon(AT)dpmms.cam.ac.uk)
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