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A088209
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Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...
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8
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1, 3, 7, 14, 28, 53, 99, 181, 327, 584, 1034, 1817, 3173, 5511, 9527, 16402, 28136, 48109, 82023, 139481, 236631, 400588, 676822, 1141489, 1921993, 3231243, 5424679, 9095126, 15230452, 25475429, 42566379, 71052157, 118489383
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OFFSET
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0,2
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COMMENTS
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Denominators form the Les Marvin sequence: A007502(n+1).
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LINKS
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FORMULA
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G.f.: (1+x-x^3)/(1-x-x^2)^2. [Corrected by Georg Fischer, Aug 16 2021]
a(n) = Fibonacci(n) + (n+1)*Fibonacci(n+1). - Paul Barry, Apr 20 2004
a(n) = a(n-1) + a(n-2) + Lucas(n). - Yuchun Ji, Apr 23 2023
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EXAMPLE
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a(3)/A007502(4) = [1;1,1,4] = 14/9.
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MATHEMATICA
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f[n_] := Numerator@ FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* Robert G. Wilson v, Mar 04 2012 *)
CoefficientList[Series[(1+x-x^3)/(-1+x+x^2)^2, {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 1, -2, -1}, {1, 3, 7, 14}, 40] (* Harvey P. Dale, Jul 13 2021 *)
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PROG
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(Haskell)
a088209 n = a088209_list !! n
a088209_list = zipWith (+) a000045_list $ tail a045925_list
(Julia) # The function 'fibrec' is defined in A354044.
n == 0 && return BigInt(1)
a, b = fibrec(n)
a + (n + 1)*b
end
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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