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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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1
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1, 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, 570655858439, 1618256772285
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OFFSET
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0,3
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COMMENTS
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In general, a(n) = (2/m)*Sum_{k=1..m-1} sin(Pi*k/m)^2(1+2*cos(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.
a(n) is the number of Motzkin n-paths of height <= 6. - Alois P. Heinz, Nov 24 2023
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LINKS
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FORMULA
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a(n) = (1/4)*Sum_{k=1..7} sin(Pi*k/8)^2*(1+2*cos(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
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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}] (* 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.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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