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
Robert G. Wilson v, Table of n, a(n) for n = 0..10000
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
KEYWORD
nonn,nice
AUTHOR
Paul D. Hanna, May 09 2003
EXTENSIONS
More terms from N. J. A. Sloane, Jul 02 2005
STATUS
approved