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Search: id:A094288
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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
1
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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
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OFFSET
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1,2
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COMMENT
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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
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a(n)=(1/4)*Sum_{k=1..7} Sin(Pi*k/8)^2(1+2Cos(Pi*k/8))^n
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MATHEMATICA
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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
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This is a different sequence from the Motzkin numbers, A001006.
Sequence in context: A005207 A094286 A094287 this_sequence A166587 A086246 A001006
Adjacent sequences: A094285 A094286 A094287 this_sequence A094289 A094290 A094291
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KEYWORD
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easy,nonn
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AUTHOR
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Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
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