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).

1, 3, 9, 404, 6355, 11482910373, 1268361281038

Offset

1,2

Links

Table of n, a(n) for n=1..7.

Peter Borwein, Stephen Choi and Frank Chu, An old conjecture of Erdos-Turán on additive bases, Math. Comp. 75 (2006), 475-484, see Table 1 p. 481.

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])={c||c=[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 200-300 solutions. - M. F. Hasler, Jul 31 2015

Crossrefs

Sequence in context: A303130 A144984 A285059 * A359615 A137036 A361389

Adjacent sequences: A260548 A260549 A260550 * A260552 A260553 A260554

Keyword

nonn,more

Author

Michel Marcus, Jul 29 2015

Extensions

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

Status

approved