%I #27 Apr 19 2018 15:02:12

%S 1,1,2,5,17,65,292,1434,7875,47098,305226,2122983,15752080,124015310,

%T 1031857395,9041908204,83186138212

%N Number of increasing sequences of Goldbach type of length n; a(0) = 1 by convention.

%C From _David S. Newman_, Feb 17 2009: (Start)

%C This sequence also arises in the following way.

%C 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.

%C Call a basis an increasing basis if its elements are arranged in increasing order, a0 < a1 < a2 < ...

%C For example, A126684: 0, 1, 2, 4, 5, 8, 10, 16, 17, 20, 21, 32, 34, 40, ... is an increasing basis.

%C 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:

%C 0

%C 0, 1

%C 0, 1, 2

%C 0, 1, 3

%C 0, 1, 2, 3

%C 0, 1, 2, 4

%C 0, 1, 2, 5

%C 0, 1, 3, 4

%C 0, 1, 3, 5

%C ...

%C How many such subsequences are there of length n? The answer is a(n+1).

%C A Goldbach sequence is then an increasing basis without the initial zero. (End)

%C The largest value for each term in any increasing basis is given by A123509. - _Martin Fuller_, Jun 01 2010

%D M. Torelli, Increasing integer sequences and Goldbach's conjecture, preprint, 1996.

%H M. Torelli, <a href="http://dx.doi.org/10.1051/ita:2006017">Increasing integer sequences and Goldbach's conjecture</a>, RAIRO - Theoretical Informatics and Applications, vol.40, no.02 (April 2006), pp.107-121.

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%o (PARI) A008932(n,pol=0)= { local(a=0, i, pol2);

%o !n && return(1);

%o i = #pol;

%o pol2 = pol^2;

%o for (i=#pol, #pol2+1,

%o a += A008932(n-1, pol+'x^i);

%o !polcoeff(pol2,i) && break;);

%o a } \\ _Martin Fuller_, Jun 01 2010

%Y Cf. A123509.

%K nonn,more

%O 0,3

%A Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it)

%E a(9)-a(14) from _Martin Fuller_, Feb 18 2009

%E Edited by _N. J. A. Sloane_, Mar 12 2009

%E a(15)-a(16) from _Sean A. Irvine_, Apr 19 2018