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A083594
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a(n) = (7 - 4*(-2)^n)/3.
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2
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1, 5, -3, 13, -19, 45, -83, 173, -339, 685, -1363, 2733, -5459, 10925, -21843, 43693, -87379, 174765, -349523, 699053, -1398099, 2796205, -5592403, 11184813, -22369619, 44739245, -89478483, 178956973, -357913939, 715827885, -1431655763, 2863311533, -5726623059
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OFFSET
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0,2
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COMMENTS
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Also generalized k-bonacci sequence a(n)=2*a(n-2)-a(n-1). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
The k-bonacci sequences are constructed using the formula a(n+k)=sum({i=1 to k-1) a(n+i) where integers a(0) to a(k-1) are given. The generalized k-bonnacci sequences are built with the formula a(n+k) =sum({i=1 to k-1}p(i)* a(n+i)), where integer coefficients p(1) to p(k-1) and integers a(0) to a(k-1) are given . The terms of such a sequence may be calculated by a formula such as: a(n>=k) = sum ({i =0 to k-1} q(i) * r(i)^n) where r(0) to r(k-1) are the roots (real or complex) of the equation x^k= sum {i=0 to i=k-1}p(i)x^i) The coefficients q(i) (real or complex) may be calculated by the system of equations: {for p=0 to k-1} sum( {(i=0 to k-1} q(i)*r(i)^p)=a(p), first given terms of the sequence For this sequence, the roots of x^2=2*x-1 are 1 and -2 The system of equations for q(0) and q(1) is q(0)+ q(1) = 1 q(0)-2*q(1)= 5 which gives q(0)=7/3 and q(1)= -4/3 and then the first proposed formula. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
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LINKS
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FORMULA
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G.f.: (1+6*x)/((1-x)*(1+2*x)).
E.g.f.: (7*exp(x)-4*exp(-2*x))/3.
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MATHEMATICA
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(7-4(-2)^Range[0, 40])/3 (* or *) LinearRecurrence[{-1, 2}, {1, 5}, 40] (* Harvey P. Dale, Feb 25 2012 *)
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CROSSREFS
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KEYWORD
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easy,sign,changed
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AUTHOR
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STATUS
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approved
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