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A192472
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Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+2).
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5
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3, 7, 15, 37, 93, 239, 619, 1611, 4203, 10981, 28713, 75115, 196563, 514463, 1346647, 3525189, 9228453, 24159415, 63248571, 165584323, 433501203, 1134914117, 2971232785, 7778770707, 20365057443, 53316366199, 139583983839, 365435492581
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OFFSET
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1,1
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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*(3*x^4-8*x^3-x^2+8*x-3)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Nov 12 2012]
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EXAMPLE
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The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^4 -> 3+4x
p(2,x)=1+x^2+x^6 -> 7+9x
p(3,x)=1+x^3+x^8 -> 15+23x
p(4,x)=1+x^4+x^10 -> 37+58x.
From these, read
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MATHEMATICA
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Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+2);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
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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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