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A192757
Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
0
1, 3, 5, 10, 17, 28, 47, 76, 125, 203, 329, 534, 865, 1400, 2267, 3668, 5937, 9607, 15545, 25154, 40700, 65856, 106558, 172415, 278975, 451392, 730368, 1181762, 1912131, 3093895, 5006028, 8099924, 13105954, 21205880, 34311835, 55517717
OFFSET
0,2
COMMENTS
The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+floor(n*r) for n>0, where p(0,x)=1 and r=(1+sqrt(5))/2. For discussions of polynomial reduction, see A192232 and A192744.
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, n_] := 1;
p[n_, x_] := x*p[n - 1, x] + Floor[(n + 1)*GoldenRatio] /; n > 0;
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}]
(* A192457 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A023611 *)
CROSSREFS
Sequence in context: A261998 A270413 A270415 * A079934 A215004 A005403
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved