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A083952
Integer coefficients a(n) of A(x), where a(n) = 1 or 2 for all n, such that A(x)^(1/2) has only integer coefficients.
29
1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
OFFSET
0,2
COMMENTS
More generally, the sequence "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. [That is true - see Theorem 17 of Heninger-Rains-Sloane (2006). - N. J. A. Sloane, Aug 27 2015] Is this sequence periodic? [It is not periodic for m = 2 or 3. Larger cases remain open. - N. J. A. Sloane, Aug 27 2015]
LINKS
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
MATHEMATICA
a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v, Nov 25 2006 *)
s = 0; a[n_] := a[n] = Block[{}, If[IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], s = s + x^n; 1, s = s + 2 x^n; 2]]; Table[ a@n, {n, 0, 104}] (* Robert G. Wilson v, Sep 08 2007 *)
CROSSREFS
Cf. A084202 (A(x)^(1/2)), A108335 (A084202 mod 4), A108336 (A084202 mod 2), A108340 (a(n) mod 2). Positions of 1's: A108783.
Sequence in context: A275308 A233138 A214708 * A245225 A214860 A263649
KEYWORD
nonn,nice
AUTHOR
Paul D. Hanna, May 09 2003
EXTENSIONS
More terms from N. J. A. Sloane, Jul 02 2005
STATUS
approved