A207187 Numbers matching polynomials y(k,x) that have x+1 as a factor; see Comments.

3, 5, 9, 13, 19, 22, 25, 27, 30, 33, 39, 43, 49, 52, 55, 59, 65, 68, 71, 83, 89, 92, 95, 101, 104, 107, 110, 113, 116, 119, 121, 124, 127, 133, 136, 139, 142, 145, 148, 151, 157, 169, 172, 175, 181, 185, 191, 194, 197, 209, 215, 218, 221, 224, 227, 230

Offset

1,1

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Links

Table of n, a(n) for n=1..56.

Example

The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Mathematica

t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* monic polynomials over {-1, 0, 1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)

Crossrefs

Cf. A206821.

Sequence in context: A078735 A212530 A004132 * A359185 A065802 A245302

Adjacent sequences: A207184 A207185 A207186 * A207188 A207189 A207190

Keyword

nonn

Author

Clark Kimberling, Feb 16 2012

Status

approved