%I #9 Oct 03 2023 22:39:22
%S 6,10,14,16,23,26,35,37,42,51,57,60,68,74,83,90,92,97,106,110,116,120,
%T 127,132,134,146,149,157,163,172,178,184,188,192,194,206,214,217,234,
%U 236,241,250,254,260,264,271,276,278,288,294,298,302,304,311
%N Numbers k matched to Zeckendorf polynomials divisible by x+1.
%C The Zeckendorf polynomials Z(x,k) are defined and ordered at A207813.
%e The first ten Zeckendorf polynomials are 1, x, x^2, x^2 + 1, x^3, x^3 + 1, x + x^3, x^4, 1 + x^4, x + x^4; their values at x=-1 are 1, -1, 1, 2, -1, 0, -2, 1, 2, 0, indicating initial terms for A207869 and this sequence.
%t fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
%t t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
%t AppendTo[fr, 1]; t = t - Fibonacci[k],
%t AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
%t {n, 1, 500}];
%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t Table[p[n, x], {n, 1, 40}]
%t Table[p[n, x] /. x -> 1, {n, 1, 120}] (* A007895 *)
%t Table[p[n, x] /. x -> 2, {n, 1, 120}] (* A003714 *)
%t Table[p[n, x] /. x -> 3, {n, 1, 120}] (* A060140 *)
%t t1 = Table[p[n, x] /. x -> -1,
%t {n, 1, 420}] (* A207869 *)
%t Flatten[Position[t1, 0]] (* this sequence *)
%t t2 = Table[p[n, x] /. x -> I, {n, 1, 420}];
%t Flatten[Position[t2, 0] (* A207871 *)
%t Denominator[Table[p[n, x] /. x -> 1/2,
%t {n, 1, 120}]] (* A207872 *)
%t Numerator[Table[p[n, x] /. x -> 1/2,
%t {n, 1, 120}]] (* A207873 *)
%Y Cf. A207813, A207869.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 21 2012
|