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A207188
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Numbers matching polynomials y(k,x) that have x as a factor; see Comments.
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5
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2, 4, 6, 9, 11, 13, 15, 17, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128
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OFFSET
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1,1
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COMMENTS
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The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.
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LINKS
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EXAMPLE
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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,...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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