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A207189 Numbers matching polynomials y(k,x) that have x-1 as a factor; see Comments. 5
5, 6, 12, 13, 15, 27, 28, 30, 34, 58, 59, 61, 65, 73, 121, 122, 124, 128, 136, 152, 248, 249, 251, 255, 263, 279, 311, 503, 504, 506, 510, 518, 534, 566, 630, 1014, 1015, 1017, 1021, 1029, 1045, 1077, 1141, 1269, 2037, 2038, 2040, 2044, 2052, 2068 (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
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: A177714 A273158 A099641 * A285404 A047319 A244368
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 16 2012
STATUS
approved

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Last modified May 25 15:42 EDT 2024. Contains 372800 sequences. (Running on oeis4.)