A207813 - OEIS

%I #14 Oct 02 2024 13:22:56

%S 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 101,103,108,111,114,119,124,129,135,137,140,150,153,161,166,169,171,

%U 177,179,182,187,195,197,205,208,211,213,218,224,229,237,239

%N Numbers that match irreducible Zeckendorf polynomials.

%C The Zeckendorf representation of a positive integer n is a unique sum

%C c(k-2)F(k) + c(k-3)F(k-1) + ... + c(1)F(3) + c(0)F(2),

%C 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 c(k-2)x^(k-2) + c(k-3)x^(k-3) + ... + c(1)x + c(0).

%C The name refers to irreducibility over the field of rational numbers.

%e n   k    Z(n)   Z(n,x)       irreducible

%e 1   2       1   1            no

%e 2   3      10   x            yes

%e 3   4     100   x^2          no

%e 4   4     101   x^2 + 1      yes

%e 5   5    1000   x^3          no

%e 6   5    1001   x^3 + 1      no

%e 7   5    1010   x^3 + x      no

%e 8   5   10000   x^4          no

%e 9   5   10001   x^4 + 1      yes

%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, 350}];

%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}] (* Zeckendorf polynomials *)

%t u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],

%t   AppendTo[u, n]], {n, 300}]; u     (* A207813 *)

%Y Cf. A206073, A206074.

%K nonn,base

%O 1,1

%A _Clark Kimberling_, Feb 20 2012