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A154691
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Expansion of (1+x+x^2) / ((1-x-x^2)*(1-x)).
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5
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1, 3, 7, 13, 23, 39, 65, 107, 175, 285, 463, 751, 1217, 1971, 3191, 5165, 8359, 13527, 21889, 35419, 57311, 92733, 150047, 242783, 392833, 635619, 1028455, 1664077, 2692535, 4356615, 7049153, 11405771, 18454927, 29860701, 48315631, 78176335
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OFFSET
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0,2
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
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FORMULA
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a(n+1) - a(n) = A006355(n+3) = A055389(n+3).
a(n) = A066629(n-1) + A066629(n).
a(n) = A006355(n+4) - 3 = A078642(n+1) - 3.
a(n+1) = a(n) + 2*A000045(n+2). - Reinhard Zumkeller, Nov 17 2013
From Colin Barker, Feb 01 2017: (Start)
a(n) = -3 + (2^(1-n)*((1-r)^n*(-2+r) + (1+r)^n*(2+r))) / r where r=sqrt(5).
a(n) = 2*a(n-1) - a(n-3) for n>2.
(End)
a(n) = 2*Fibonacci(n+3) - 3. - Greg Dresden, Oct 10 2020
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MAPLE
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A154691 := proc(n) coeftayl( (1+x+x^2)/(1-x-x^2)/(1-x), x=0, n) ; end proc:
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MATHEMATICA
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Fibonacci[Range[3, 5! ]]*2-3 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)
CoefficientList[Series[(1 + x + x^2)/((1 - x - x^2)(1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 18 2012 *)
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PROG
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(Haskell)
a154691 n = a154691_list !! n
a154691_list = 1 : zipWith (+)
a154691_list (drop 2 $ map (* 2) a000045_list)
-- Reinhard Zumkeller, Nov 17 2013
(PARI) Vec((1+x+x^2) / ((1-x-x^2)*(1-x)) + O(x^60)) \\ Colin Barker, Feb 01 2017
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CROSSREFS
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Cf. A000045, A006355, A055389, A066629, A078642.
Cf. A001595, A166863.
Sequence in context: A136851 A155339 A122886 * A306902 A227121 A078447
Adjacent sequences: A154688 A154689 A154690 * A154692 A154693 A154694
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KEYWORD
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easy,nonn
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AUTHOR
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R. J. Mathar, Jan 14 2009
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STATUS
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approved
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