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 A008932 Number of increasing sequences of Goldbach type of length n; a(0) = 1 by convention. 11
 1, 1, 2, 5, 17, 65, 292, 1434, 7875, 47098, 305226, 2122983, 15752080, 124015310, 1031857395 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 REFERENCES M. Torelli, Increasing integer sequences and Goldbach's conjecture, preprint, 1996. LINKS M. Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications, vol.40, no.02 (April 2006), pp.107-121. 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(n-1, pol+'x^i); !polcoeff(pol2, i) && break; ); a } \\ Martin Fuller, Jun 01 2010 CROSSREFS Cf. A123509. Sequence in context: A150013 A123166 A052539 * A167809 A262449 A062881 Adjacent sequences:  A008929 A008930 A008931 * A008933 A008934 A008935 KEYWORD nonn,more AUTHOR Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it) EXTENSIONS a(9)-a(14) from Martin Fuller, Feb 18 2009 Edited by N. J. A. Sloane, Mar 12 2009 STATUS approved

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