

A260551


Number of polynomials P = Sum_{k=0..m} x^{d(k)} with 0 = d(0) < ... < d(m) and P^2 = Sum_{k>=0} B(k) x^k such that B(k) <= n for all k and B(k) > 0 for k <= d(m).


3




OFFSET

1,2


LINKS



EXAMPLE

For n=1, the only possible polynomial is P = 1 (the coefficient of x^0 must always be 1), and its square P^2 = 1 satisfies the conditions. If another term is added, there will be a coefficient 2 > n in the square, which is forbidden.
For n=2, the 3 polynomials are {1, x+1, x^3+x+1}. P = x^2+1 is excluded because P^2 has a zero coefficient for x^1. P = x^2+x+1 is excluded because P^2 has a coefficient 3 > n which is forbidden. If the degree is > 3, then either there will be a zero coefficient in P^2 below deg(P), or there will be a coefficient > 2.
For n=3, the 9 polynomials are {1, x+1, x^2+x+1, x^3+x+1, x^4+x^2+x+1, x^5+x^2+x+1, x^5+x^3+x+1, x^7+x^4+x^2+x+1, x^8+x^5+x^2+x+1}.


PROG

(PARI) A260551(n, c=1, L=2<<[1, 3, 8, 40, 52, 264, 328][n])={cc=[1]; forstep(i=2, L, 2, normlp(P2=Pol(binary(1+i))^2)>n&&next; for(k=1, #binary(i), component(P2, k)next(2)); if(type(c)!="t_INT", c=concat(c, Pol(binary(1+i))), c++)); c} \\ Use 2nd arg=0 or [] to get the list of polynomials. For n>3 this code takes too long, but you may give a lower limit as 3rd arg to get quickly a list of the first 200300 solutions.  M. F. Hasler, Jul 31 2015


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS

Definition and examples clarified by M. F. Hasler, Jul 31 2015


STATUS

approved



