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A192964
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Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
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3
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1, 0, 3, 7, 16, 31, 57, 100, 171, 287, 476, 783, 1281, 2088, 3395, 5511, 8936, 14479, 23449, 37964, 61451, 99455, 160948, 260447, 421441, 681936, 1103427, 1785415, 2888896, 4674367, 7563321, 12237748, 19801131, 32038943, 51840140, 83879151
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OFFSET
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0,3
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COMMENTS
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The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)-n+n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.
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LINKS
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Table of n, a(n) for n=0..35.
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FORMULA
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a(n)=3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4).
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MATHEMATICA
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q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^2 - n;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
(* A192964 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192965 *)
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CROSSREFS
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Cf. A192232, A192744, A192951, A192965.
Sequence in context: A184677 A224340 A000412 * A179904 A161810 A084631
Adjacent sequences: A192961 A192962 A192963 * A192965 A192966 A192967
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, Jul 13 2011
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STATUS
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approved
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