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A207813 Numbers that match irreducible Zeckendorf polynomials.

%I A207813 #14 Oct 02 2024 13:22:56
%S A207813 2,4,9,17,19,25,27,30,40,43,46,53,56,59,61,67,69,72,77,82,85,93,95,98,
%T A207813 101,103,108,111,114,119,124,129,135,137,140,150,153,161,166,169,171,
%U A207813 177,179,182,187,195,197,205,208,211,213,218,224,229,237,239
%N A207813 Numbers that match irreducible Zeckendorf polynomials.
%C A207813 The Zeckendorf representation of a positive integer n is a unique sum
%C A207813 c(k-2)F(k) + c(k-3)F(k-1) + ... + c(1)F(3) + c(0)F(2),
%C A207813 where F=A000045 (Fibonacci numbers), c(k-2)=1, and for j=0,1,...,k-3, there are two restrictions on coefficients: c(j) is 0 or 1, and c(j)c(j+1)=0; viz., no two consecutive Fibonacci numbers appear. The Zeckendorf polynomial Z(n,x) is introduced here as
%C A207813 c(k-2)x^(k-2) + c(k-3)x^(k-3) + ... + c(1)x + c(0).
%C A207813 The name refers to irreducibility over the field of rational numbers.
%e A207813 n   k    Z(n)   Z(n,x)       irreducible
%e A207813 1   2       1   1            no
%e A207813 2   3      10   x            yes
%e A207813 3   4     100   x^2          no
%e A207813 4   4     101   x^2 + 1      yes
%e A207813 5   5    1000   x^3          no
%e A207813 6   5    1001   x^3 + 1      no
%e A207813 7   5    1010   x^3 + x      no
%e A207813 8   5   10000   x^4          no
%e A207813 9   5   10001   x^4 + 1      yes
%t A207813 fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
%t A207813  t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
%t A207813  AppendTo[fr, 1]; t = t - Fibonacci[k],
%t A207813  AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
%t A207813      {n, 1, 350}];
%t A207813 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207813 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A207813 Table[p[n, x], {n, 1, 40}] (* Zeckendorf polynomials *)
%t A207813 u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
%t A207813   AppendTo[u, n]], {n, 300}]; u     (* A207813 *)
%Y A207813 Cf. A206073, A206074.
%K A207813 nonn,base
%O A207813 1,1
%A A207813 _Clark Kimberling_, Feb 20 2012