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A207190 Numbers matching polynomials y(k,x) that have x^2 + 1 as a factor; see Comments.

%I A207190 #6 Mar 30 2012 18:58:12
%S A207190 8,20,25,27,37,45,55,59,64,79,96,101,116,124,134,164,184,194,199,204,
%T A207190 209,214,224,239,244,255,260,275,320,335,340,376,381,396,406,411,416,
%U A207190 421,426,436,441,456,461,471,481,486,491,496,501,503,513,518
%N A207190 Numbers matching polynomials y(k,x) that have x^2 + 1 as a factor; see Comments.
%C A207190 The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.
%e A207190 The first 13 polynomials:
%e A207190 1 .... 1
%e A207190 2 .... x
%e A207190 3 .... x + 1
%e A207190 4 .... x^2
%e A207190 5 .... x^2 - 1
%e A207190 6 .... x^2 - x
%e A207190 7 .... x^2 - x - 1
%e A207190 8 .... x^2 + 1
%e A207190 9 .... x^2 + x
%e A207190 10 ... x^2 + x + 1
%e A207190 11 ... x^3
%e A207190 12 ... x^3 - 1
%e A207190 13 ... x^3 - x
%e A207190 The list exemplifies these sequences:
%e A207190 A207187 (multiples of x + 1): 3,5,9,13,...
%e A207190 A207188 (multiples of x): 2,4,6,9,11,13,...
%e A207190 A207189 (multiples of x - 1): 5,6,12,13,...
%e A207190 A207190 (multiples of x^2 + 1): 8,20,25,27,...
%t A207190 t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
%t A207190 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207190 p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
%t A207190 TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
%t A207190 f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
%t A207190 q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
%t A207190 q2[n_] := p[n - f[k] + 2]
%t A207190 y1 = Table[p[n], {n, 1, 4}];
%t A207190 Do[AppendTo[y1,
%t A207190   Join[Table[q1[n], {n, f[k], g[k] - 1}],
%t A207190    Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
%t A207190 y = Flatten[y1]; (* polynomials over {-1,0,1} *)
%t A207190 TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
%t A207190 Table[y[[n]] /. x -> -1, {n, 1, 300}];
%t A207190 Flatten[Position[%, 0]]  (* A207187 *)
%t A207190 Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
%t A207190 Flatten[Position[%, 0]]  (* A207188 *)
%t A207190 Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
%t A207190 Flatten[Position[%, 0]]  (* A207189 *)
%t A207190 Table[y[[n]] /. x -> I, {n, 1, 600}] ;
%t A207190 Flatten[Position[%, 0]]  (* A207190 *)
%Y A207190 Cf. A206821.
%K A207190 nonn
%O A207190 1,1
%A A207190 _Clark Kimberling_, Feb 16 2012