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A092387
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a(n) = Fibonacci(2*n+1) + Fibonacci(2*n-1) + 2.
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2
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5, 9, 20, 49, 125, 324, 845, 2209, 5780, 15129, 39605, 103684, 271445, 710649, 1860500, 4870849, 12752045, 33385284, 87403805, 228826129, 599074580, 1568397609, 4106118245, 10749957124, 28143753125, 73681302249, 192900153620
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OFFSET
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1,1
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COMMENTS
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Let b(k)=sum(i=1,k,F(2*n*i)*binomial(k,i)) where F(k) denotes the k-th Fibonacci number. The (b(k)) sequence satisfies the recursion: b(k)=a(n)*(b(k-1)-b(k-2)).
Same as the number of Kekulé structures in polyphenanthrene in terms of the number of hexagons. - Parthasarathy Nambi, Aug 22 2006
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LINKS
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FORMULA
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a(1)=5, a(2)=9, a(3)=20, a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
a(n) = 3 + floor((1+phi)^n) where phi = (1+sqrt(5))/2.
G.f.: -x*(5 - 11*x + 4*x^2)/((x-1)(x^2 - 3*x + 1)).
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MATHEMATICA
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CoefficientList[Series[-(5-11*x+4*x^2)/((x-1)(x^2-3*x+1)), {x, 0, 30}], x] (* Vincenzo Librandi, May 06 2012 *)
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PROG
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(Magma) I:=[5, 9, 20]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 06 2012
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Better definition from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 20 2004
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STATUS
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approved
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