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COMMENTS
| Comment from David 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 ordered in the library ordering, 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)
Comment from Martin Fuller: The largest value for each term in any increasing basis is given by A123509.
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PROG
| (PARI code from Martin Fuller) A008932(n, pol=0)= { local(a=0, i, pol2);
!n & return(1);
i = #pol;
pol2 = pol^2;
for (i=#pol, #pol2+1,
a += A008932(n-1, pol+'x^i);
!polcoeff(pol2, i) & break; );
a }
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