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A192313
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Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.
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3
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1, 2, 5, 13, 34, 91, 247, 680, 1893, 5319, 15056, 42867, 122605, 351898, 1012729, 2920521, 8435362, 24392655, 70599403, 204472264
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OFFSET
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1,2
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COMMENTS
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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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FORMULA
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Empirical G.f.: x*(x+1)*(x^2-3*x+1)/(x^4+6*x^3+x^2-4*x+1). [Colin Barker, Nov 13 2012]
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EXAMPLE
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The first five polynomials at A157751 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=2+x -> 2+x
p2(x)=4+2x+x^2 -> 5+3x
p3(x)=8+4x+4x^2+x^3 -> 13+10x
p4(x)=16+8x+12x^2+4x^3+x^4 -> 34+31x.
From these, we read
A192313=(1,2,5,13,34,...) and A192314=(0,1,3,19,31,...)
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MATHEMATICA
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q[x_] := x + 1;
p[0, x_] := 1;
p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /; n > 0 (* A157751 *)
Table[Simplify[p[n, x]], {n, 0, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 20}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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