
COMMENTS

From David S. Newman, Feb 17 2009: (Start)
This sequence also arises in the following way.
Call a set A of nonnegative integers a basis if every nonnegative integer can be written as the sum of two (not necessarily distinct) elements of A.
Call a basis an increasing basis if its elements are arranged in increasing order, a0 < a1 < a2 < ...
For example, A126684: 0, 1, 2, 4, 5, 8, 10, 16, 17, 20, 21, 32, 34, 40, ... is an increasing basis.
Now consider the set of all initial subsequences of any length {a0, a1, a2,...,an} of all the increasing bases. These can be arranged in lexicographic order, giving:
0
0, 1
0, 1, 2
0, 1, 3
0, 1, 2, 3
0, 1, 2, 4
0, 1, 2, 5
0, 1, 3, 4
0, 1, 3, 5
...
How many such subsequences are there of length n? The answer is a(n+1).
A Goldbach sequence is then an increasing basis without the initial zero. (End)
The largest value for each term in any increasing basis is given by A123509.  Martin Fuller, Jun 01 2010


PROG

(PARI) A008932(n, pol=0)= { local(a=0, i, pol2);
!n && return(1);
i = #pol;
pol2 = pol^2;
for (i=#pol, #pol2+1,
a += A008932(n1, pol+'x^i);
!polcoeff(pol2, i) && break; );
a } \\ Martin Fuller, Jun 01 2010
