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A192143
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0-sequence of reduction of hexagonal numbers sequence by x^2 -> x+1.
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2
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1, 1, 16, 44, 134, 332, 787, 1747, 3736, 7726, 15580, 30760, 59685, 114117, 215472, 402464, 744674, 1366484, 2489175, 4504695, 8104536, 14504226, 25833336, 45811344, 80916169, 142400137, 249760912, 436706132, 761385086, 1323910556
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OFFSET
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1,3
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COMMENTS
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See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
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LINKS
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FORMULA
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Empirical G.f.: x*(1-3*x+15*x^2-12*x^3+6*x^4)/(1-x)/(1-x-x^2)^3. [Colin Barker, Feb 11 2012]
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MATHEMATICA
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c[n_] := n (2 n - 1); (* hexagonal numbers, A000384 *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
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,
30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192143 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192144 *)
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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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