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A192431
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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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2
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0, 1, 4, 15, 52, 185, 648, 2287, 8040, 28321, 99660, 350879, 1235036, 4347705, 15304208, 53873695, 189642192, 667570433, 2349942420, 8272149359, 29119170180, 102503781241, 360828342424, 1270168882575, 4471181087032, 15739215003425
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OFFSET
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0,3
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COMMENTS
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The polynomial p(n,x) is defined by (u^n+v^n)//2)^n+(u^n-v^n)/(2d), where u=x+d, v=x-d, d=sqrt(x^2+2). For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
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LINKS
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Table of n, a(n) for n=0..25.
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FORMULA
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Conjectures from Colin Barker, Jun 07 2019: (Start)
G.f.: x*(1 + x)^2 / (1 - 2*x - 6*x^2 + 2*x^3 + x^4).
a(n) = 2*a(n-1) + 6*a(n-2) - 2*a(n-3) - a(n-4) for n>3.
(End)
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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)=1+x -> 1+x
p(2,x)=2+3x+x^2 -> 3+4x
p(3,x)=2+7x+6x^2+x^3 -> 9+15x
p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x.
From these, read
A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...)
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MATHEMATICA
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(See A192430.)
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CROSSREFS
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Cf. A192232, A192430.
Sequence in context: A027853 A132894 A117917 * A329253 A161125 A027295
Adjacent sequences: A192428 A192429 A192430 * A192432 A192433 A192434
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KEYWORD
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nonn
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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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