A207869 a(n) = Z(n,-1), where Z(n,x) is the n-th Zeckendorf polynomial.
1, -1, 1, 2, -1, 0, -2, 1, 2, 0, 2, 3, -1, 0, -2, 0, 1, -2, -1, -3, 1, 2, 0, 2, 3, 0, 1, -1, 2, 3, 1, 3, 4, -1, 0, -2, 0, 1, -2, -1, -3, 0, 1, -1, 1, 2, -2, -1, -3, -1, 0, -3, -2, -4, 1, 2, 0, 2, 3, 0, 1, -1, 2, 3, 1, 3, 4, 0, 1, -1, 1, 2, -1, 0, -2, 2, 3, 1, 3, 4, 1, 2, 0
Offset: 1
Keywords
Examples
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 A207870.
Programs
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Mathematica
fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n], {n, 1, 500}]; b[n_] := Reverse[Table[x^k, {k, 0, n}]] p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]] Table[p[n, x], {n, 1, 40}] Table[p[n, x] /. x -> 1, {n, 1, 120}] (* A007895 *) Table[p[n, x] /. x -> 2, {n, 1, 120}] (* A003714 *) Table[p[n, x] /. x -> 3, {n, 1, 120}] (* A060140 *) t1 = Table[p[n, x] /. x -> -1, {n, 1, 420}] (* A207869 *) Flatten[Position[t1, 0]] (* A207870 *) t2 = Table[p[n, x] /. x -> I, {n, 1, 420}]; Flatten[Position[t2, 0]] (* A207871 *) Denominator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]] (* A207872 *) Numerator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]] (* A207873 *)
Comments