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A094309
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Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 5.
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1
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1, 4, 14, 44, 132, 384, 1096, 3088, 8624, 23936, 66144, 182208, 500800, 1374208, 3766400, 10313984, 28226304, 77211648, 211138048, 577223680, 1577772032, 4312088576, 11783915520, 32200396800, 87985401856, 240405151744
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| In general a(n,m,j,k)=2/m*Sum_{r=1..m-1} Sin(j*r*Pi/m)Sin(k*r*Pi/m)(1+2Cos(Pi*r/m))^n) is the number of (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) = j, s(n) = k.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (4,-2,-4).
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FORMULA
| a(n)=(1/3)*Sum_{k=1..5} Sin(Pi*k/3)Sin(5Pi*k/6)(1+2Cos(Pi*k/6))^n
G.f. : x^3/((1-2x)(1-2x-2x^2)); a(n)=4a(n-1)-2a(n-2)-4a(n-3); a(n)=(1+sqrt(3))^n(3/2+5sqrt(3)/6)+(1-sqrt(3))^n(3/2-5sqrt(3)/6)-2^(n+1) [offset 0]. - Paul Barry (pbarry(AT)wit.ie), Jul 28 2004
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MATHEMATICA
| f[n_] := FullSimplify[ TrigToExp[(1/3)Sum[ Sin[Pi*k/3] Sin[5Pi*k/6](1 + 2Cos[Pi*k/6])^n, {k, 1, 5}]]]; Table[ f[n], {n, 3, 28}] (from Robert G. Wilson v 2004)
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CROSSREFS
| Sequence in context: A062109 A118042 A006645 * A000300 A005323 A027831
Adjacent sequences: A094306 A094307 A094308 * A094310 A094311 A094312
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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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