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A094288
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Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 1.
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0
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1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113633, 310557, 853333, 2355861, 6531062, 18171848, 50722229, 141973073, 398351055, 1120056347, 3155043447, 8901325751, 25147423616, 71127785002, 201381834019
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| In general a(n)=2/m*Sum_{k=1..m-1} Sin(Pi*k/m)^2(1+2Cos(Pi*k/m))^n counts the (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 1. Here is m=8.
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FORMULA
| a(n)=(1/4)*Sum_{k=1..7} Sin(Pi*k/8)^2(1+2Cos(Pi*k/8))^n.
Conjecture: a(n)= +7*a(n-1) -15*a(n-2) +5*a(n-3) +15*a(n-4) -9*a(n-5) -3*a(n-6) +a(n-7) with G.f. -x*(1-5*x+5*x^2+6*x^3-7*x^4-2*x^5+x^6) / ( (x-1)*(x^2+2*x-1)*(x^4-4*x^3-2*x^2+4*x-1) ). - R. J. Mathar, Dec 20 2011
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MATHEMATICA
| f[n_] := FullSimplify[ TrigToExp[(1/4)*Sum[Sin[Pi*k/8]^2(1 + 2Cos[Pi*k/8])^n, {k, 1, 7}]]]; Table[ f[n], {n, 28}] (from Robert G. Wilson v Jun 18 2004)
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CROSSREFS
| This is a different sequence from the Motzkin numbers, A001006.
Sequence in context: A005207 A094286 A094287 * A168051 A166587 A086246
Adjacent sequences: A094285 A094286 A094287 * A094289 A094290 A094291
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KEYWORD
| easy,nonn
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AUTHOR
| Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
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