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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, 9041908204, 83186138212 (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 A052539 A123166 * A167809 A262449 A346506
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
a(15)-a(16) from Sean A. Irvine, Apr 19 2018
STATUS
approved

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Last modified March 19 06:05 EDT 2024. Contains 370952 sequences. (Running on oeis4.)