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Search: id:A094292
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| A094292 |
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Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 4. |
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+0
1
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| 1, 3, 9, 25, 68, 182, 483, 1275, 3355, 8811, 23112, 60580, 158717, 415715, 1088661, 2850645, 7463884, 19541994, 51163695, 133951675, 350695511, 918141623, 2403740304, 6293097000, 16475579353, 43133687427, 112925557953
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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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.
a(n+1) is an inverse Catalan transform of F(3n)/F(3). The g.f. may be obtained from that of A001076 under the mapping g(x)-> g(x(1-x)). - Paul Barry (pbarry(AT)wit.ie), Nov 17 2004
A transform of Fib(2n) : Fib(2n) may be recovered as sum{k=0..2n, sum{j=0..k, C(0,2n-k)C(k,j)(-1)^(k-j)*A094292(j)}}. - Paul Barry (pbarry(AT)wit.ie), Jun 10 2005
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FORMULA
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a(n)=(2/5)*Sum(k, 1, 4, Sin(2Pi*k/5)Sin(4Pi*k/5)(1+2Cos(Pi*k/5))^n)
a(n)=4a(n-1)-3a(n-2)-2a(n-3)+a(n-4) G.f.: (x^2-x^3)/(1-4x+3x^2+2x^3-x^4) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 16 2004
a(n) = (Fibonacci(2*n)-Fibonacci(n))/2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 17 2004
a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*F(3n-3k)/F(3)} - Paul Barry (pbarry(AT)wit.ie), Nov 17 2004
a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(2k)} - Paul Barry (pbarry(AT)wit.ie), Jun 10 2005
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PROGRAM
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(Mupad)(numlib::fibonacci(2*n)-numlib::fibonacci(n))/2 $ n = 2..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008
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CROSSREFS
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Sequence in context: A106514 A085327 A069403 this_sequence A000242 A077846 A005322
Adjacent sequences: A094289 A094290 A094291 this_sequence A094293 A094294 A094295
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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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