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A102714
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Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).
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1
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2, 5, 14, 36, 95, 248, 650, 1701, 4454, 11660, 30527, 79920, 209234, 547781, 1434110, 3754548, 9829535, 25734056, 67372634, 176383845, 461778902, 1208952860, 3165079679, 8286286176, 21693778850, 56795050373, 148691372270, 389279066436, 1019145827039
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OFFSET
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0,1
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COMMENTS
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A floretion-generated sequence relating Fibonacci numbers.
Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 2dia[I]forseq[ + .5'i + .5'ii' + .5'ij' + .5'ik' ], 2dia[J]forseq = 2dia[K]forseq = A001654, mixforseq = A001519, tesforseq = A099016, vesforseq = A000004. Identity used: dia[I] + dia[J] + dia[K] + mix + tes = ves
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), a(0) = 2, a(1) = 5, a(2) = 14.
a(n) + 2*a(n+1) + a(n+2) = A055849(n+2).
a(n) = (2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5. - Colin Barker, Oct 01 2016
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MATHEMATICA
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CoefficientList[Series[(x+2)/((x+1)(x^2-3x+1)), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, 2, -1}, {2, 5, 14}, 30] (* Harvey P. Dale, Apr 22 2012 *)
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PROG
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(PARI) a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016
(PARI) Vec((x+2)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Corrected by T. D. Noe, Nov 02 2006, Nov 07 2006
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STATUS
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approved
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