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A207187 Numbers matching polynomials y(k,x) that have x+1 as a factor; see Comments. 5
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 (list; graph; refs; listen; history; text; internal format)
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 * A065802 A245302 A118028

Adjacent sequences:  A207184 A207185 A207186 * A207188 A207189 A207190

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 16 2012

STATUS

approved

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Last modified November 16 22:05 EST 2019. Contains 329208 sequences. (Running on oeis4.)