A192757 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

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.

Crossrefs

Cf. A192744, A192232.

Programs

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 *)