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%I #14 Dec 02 2021 13:56:06
%S 1,2,4,8,16,28,57,114,221,451,893,1792,3549,7104,14212,28445,56894,
%T 113792,227554,455124,910208,1820449,3640907,7281813,14563613,
%U 29127251,58254501,116508984,233017889,466035877,932071736
%N a(0) = A002858(1) = 1, followed by the greatest Ulam numbers A002858 to form a complete sequence (see algorithm below).
%C This sequence starts at a(0)=1, subsequent terms a(n) for n>0 being obtained by selecting the (greatest Ulam number) <= 1+Sum_{i=0..n-1} a(i). This ensures that the sequence is complete because Sum_{i=0..n-1} a(i) >= a(n)-1, for all n>=0 and a(0)=1, is a necessary and sufficient condition for completeness.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_number">Ulam number</a>.
%F a(n) = (greatest Ulam number) <= 1+Sum_{i=0..n-1} a(i), with a(0) = 1.
%e Given that the first 7 terms of the sequence are 1, 2, ..., 28, 57 then a(7)=(greatest Ulam number) <= (1+2+...+28, 57) + 1 = 117, hence a(7)=114.
%t lst1 = Last/@ReadList["https://oeis.org/A002858/b002858.txt", {Number, Number}]; lst={1, 2}; n=3; Do[s=Total@lst; While[s+1>=lst1[[n]], n++]; AppendTo[lst, lst1[[n-1]]], 16]; lst
%Y Cf. A002858.
%K nonn,more
%O 0,2
%A _Frank M Jackson_, Oct 17 2021
%E a(18)-a(30) from _Amiram Eldar_, Oct 17 2021
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