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A114849
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F(4n+4)/F(4)-F(3n+3)/F(3) where F(n)=A000045(n).
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0
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0, 3, 31, 257, 1950, 14164, 100464, 702919, 4878575, 33695365, 232040622, 1595043816, 10952137040, 75149854091, 515435467055, 3534332855753, 24230970910510, 166108203507452, 1138635489987488, 7804802111777935
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OFFSET
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0,2
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COMMENTS
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The limit as n -> infinity of a(n+1)/a(n) is (1+sqrt(5))*(2+sqrt(5))/2 = 6.8541019662...
Old name was: Difference between two Fibonacci cycles A000045 (three's cycle minus two's cycle).
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LINKS
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FORMULA
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a(n) = F(4n+4)/F(4)-F(3n+3)/F(3) = (2*F(4n+4)-3*F(3n+3))/6, where F=A000045.
G.f.: x*(2*x-3) / ((x^2-7*x+1)*(x^2+4*x-1)). - Colin Barker, Mar 15 2013
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MATHEMATICA
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a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] b = Table[a[4*(n + 1)]/a[4], {n, 0, 25}]; c = Table[a[3*(n + 1)]/a[3], {n, 0, 25}]; aout = b - c
LinearRecurrence[{11, -28, -3, 1}, {0, 3, 31, 257}, 20] (* Harvey P. Dale, Jun 09 2022 *)
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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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