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A192474 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^(n+1)+x^(2n). 2
3, 4, 8, 17, 40, 98, 247, 632, 1632, 4237, 11036, 28802, 75259, 196796, 514840, 1347257, 3526176, 9230050, 24161999, 63252752, 165591088, 433512149, 1134931828, 2971261442, 7778817075, 20365132468, 53316487592, 139584180257, 365435810392 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
LINKS
FORMULA
Empirical G.f.: -x*(2*x^4-2*x^3-9*x^2+11*x-3)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Nov 12 2012]
EXAMPLE
The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+2x^2 -> 3+2x
p(2,x)=1+x^3+x^4 -> 4+5x
p(3,x)=1+x^4+x^6 -> 8+11x
p(4,x)=1+x^5+x^8 -> 17+26x.
From these, read
A192474=(3,4,8,17,...) and A192475=(2,5,11,26,...)
MATHEMATICA
q[x_] := x + 1;
p[n_, x_] := 1 + x^(n + 1) + x^(2 n);
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}]
(* A192474 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192475 *)
CROSSREFS
Sequence in context: A198633 A153057 A215095 * A183494 A107429 A061273
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 01 2011
STATUS
approved

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Last modified March 5 09:09 EST 2024. Contains 370545 sequences. (Running on oeis4.)