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A062114
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a(n) = 2*Fibonacci(n) - (1 - (-1)^n)/2.
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11
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0, 1, 2, 3, 6, 9, 16, 25, 42, 67, 110, 177, 288, 465, 754, 1219, 1974, 3193, 5168, 8361, 13530, 21891, 35422, 57313, 92736, 150049, 242786, 392835, 635622, 1028457, 1664080, 2692537, 4356618, 7049155, 11405774, 18454929, 29860704, 48315633, 78176338, 126491971
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OFFSET
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0,3
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LINKS
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FORMULA
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A bistable recurrence; Fibonacci with a grain of salt: a(0)=0; a(1)=1; a(n) = a(n-1) + a(n-2) + (1 + (-1)^n)/2.
a(n+1) = Sum_{k=0..n} binomial(n-floor(k/2), floor(k/2)). - Benoit Cloitre, May 05 2005
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4), n > 3. - Harvey P. Dale, Nov 02 2011
G.f.: x*(1+x-x^2)/( (1-x)*(1+x)*(1-x-x^2) ). - R. J. Mathar, Aug 12 2012
a(n) = -(-1)^n * a(-n) for all n in Z. - Michael Somos, Oct 17 2018
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EXAMPLE
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a(4) = a(3) + a(2) + (1+1)/2 = 3 + 2 + 1 = 6.
G.f. = x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 25*x^7 + ... - Michael Somos, Oct 17 2018
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MAPLE
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2*combinat[fibonacci](n)-(1-(-1)^n)/2 ;
# second Maple program:
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-1|2|1>>^n.<<[$0..3][]>>)[1$2]:
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MATHEMATICA
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Table[2Fibonacci[n]-(1-(-1)^n)/2, {n, 0, 40}] (* or *) LinearRecurrence[ {1, 2, -1, -1}, {0, 1, 2, 3}, 41] (* Harvey P. Dale, Nov 02 2011 *)
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PROG
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(PARI) { h=-1; g=1; for (n=0, 400, f=g + h; h=g; g=f; write("b062114.txt", n, " ", 2*f - (1 - (-1)^n)/2) ) } \\ Harry J. Smith, Aug 01 2009
(PARI) x='x+O('x^30); concat([0], Vec(x*(1+x-x^2)/((1-x)*(1+x)*(1-x-x^2) ))) \\ G. C. Greubel, Oct 16 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x-x^2)/((1-x)*(1+x)*(1-x-x^2)))); // G. C. Greubel, Oct 16 2018
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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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STATUS
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approved
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