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A136376
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a(n) = n*F(n) + (n-1)*F(n-1).
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6
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1, 3, 8, 18, 37, 73, 139, 259, 474, 856, 1529, 2707, 4757, 8307, 14428, 24942, 42941, 73661, 125951, 214739, 365166, 619508, 1048753, 1771943, 2988457, 5031843, 8459504, 14201994, 23811349, 39873841, 66695539, 111440227, 186016962
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OFFSET
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1,2
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COMMENTS
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For n>2, mod 2 = (0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, ...), i.e., two evens followed by four odds (repeating).
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LINKS
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FORMULA
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a(n) = n*F(n) + (n-1)*F(n-1). Equals the matrix product A128064 (unsigned) * A000045.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).
G.f.: x*(1+x)*(1+x^2)/(x^2+x-1)^2. (End)
a(n) = ((n+1)*F(n)+(n-1)*L(n))/2, where L(n) are Lucas numbers (A000032).
E.g.f.: (exp(phi*x)*(phi^3*x-1)-exp(-x/phi)*(phi^3+x)/phi)/(sqrt(5)*phi)+1, where phi=(1+sqrt(5))/2.
(End)
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EXAMPLE
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a(5) = 37 = a(n)*F(n) + (n-1)*F(n-1) = 5*5 + 4*3 = 25 + 12.
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MATHEMATICA
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Table[n*Fibonacci[n] + (n - 1)*Fibonacci[n - 1], {n, 1, 50}] (* Stefan Steinerberger, Dec 28 2007 *)
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PROG
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(PARI) Vec(x*(1+x)*(1+x^2)/(x^2+x-1)^2 + O(x^100)) \\ Altug Alkan, Oct 28 2015
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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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STATUS
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approved
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