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A192387
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Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
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3
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0, 2, 4, 32, 96, 512, 1856, 8576, 33792, 147456, 602112, 2566144, 10637312, 44892160, 187269120, 787087360, 3292069888, 13812760576, 57837355008, 242497880064, 1015868817408, 4258009186304, 17841063460864, 74771320537088, 313317428035584
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OFFSET
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1,2
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COMMENTS
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The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+5). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
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LINKS
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Table of n, a(n) for n=1..25.
Index entries for linear recurrences with constant coefficients, signature (2,12,-8,-16).
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FORMULA
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a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: 2*x^2 / (16*x^4+8*x^3-12*x^2-2*x+1). [Colin Barker, Dec 09 2012]
a(n) = 2^n*A112576(n). - R. J. Mathar, Mar 08 2021
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EXAMPLE
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The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=3+x+3x^2 -> 8+4x
p(3,x)=12x+4x^2+4x^3 -> 8+32x
p(4,x)=9+6x+31x^2+10x^3+5x^4 -> 96+96x.
From these, read
A192386=(1,0,8,8,96,...) and A192387=(0,2,4,32,96,...)
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MATHEMATICA
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(See A192386.)
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CROSSREFS
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Cf. A192232, A192386, A192388.
Sequence in context: A101575 A197099 A009098 * A320624 A133127 A122693
Adjacent sequences: A192384 A192385 A192386 * A192388 A192389 A192390
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Jun 30 2011
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STATUS
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approved
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