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A207190 Numbers matching polynomials y(k,x) that have x^2 + 1 as a factor; see Comments. 5
8, 20, 25, 27, 37, 45, 55, 59, 64, 79, 96, 101, 116, 124, 134, 164, 184, 194, 199, 204, 209, 214, 224, 239, 244, 255, 260, 275, 320, 335, 340, 376, 381, 396, 406, 411, 416, 421, 426, 436, 441, 456, 461, 471, 481, 486, 491, 496, 501, 503, 513, 518 (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..52.

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]; (* 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: A100212 A083094 A164916 * A110116 A295971 A317760

Adjacent sequences:  A207187 A207188 A207189 * A207191 A207192 A207193

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 16 2012

STATUS

approved

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Last modified December 6 14:44 EST 2019. Contains 329806 sequences. (Running on oeis4.)