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A331809 a(1) = 1; a(2) = 2; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2*a(i) nor Sum_{j=1..n-1} ( b_j*a(j) ) where 0 < i < n, b_j = 1 or 0.

%I A331809 #45 Feb 16 2025 08:33:59
%S A331809 1,2,5,9,13,31,35,92,118,280,350,866,1102,2668,3368,8240,10444,25420,
%T A331809 32156,78464,99352,242128,306440,747272,945976,2306128,2919008,
%U A331809 7117088,9009040,21964144,27802160,67784384,85802464,209191168,264795488,645591584,817196512,1992379072
%N A331809 a(1) = 1; a(2) = 2; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2*a(i) nor Sum_{j=1..n-1} ( b_j*a(j) ) where 0 < i < n, b_j = 1 or 0.
%C A331809 Inserting the additional term a(0) = 3 would result in a so-called complete sequence after sorting. (The sorted sequence will then meet Brown's criterion.)
%H A331809 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrownsCriterion.html">Brown's Criterion</a>
%H A331809 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteSequence.html">Complete Sequence</a>
%o A331809 (PARI) /* a(n) for 0<n */
%o A331809 upto(lim)={my(a=[1, 2], b=[]); for(i=1, lim, forsubset(#a, x, b=concat(b, [vecsum(vecextract(a, x))])); b=setminus(vecsort(b, , 8), a); for(j=1, #a, b=concat(b, [2*a[j]]); b=vecsort(b, , 8)); if(setsearch(b, i)==0, a=concat(a, [i]); a=vecsort(a, , 8)) ); a}
%o A331809 { upto(200) }
%Y A331809 Cf. A126326, A331800, A331811.
%K A331809 nonn
%O A331809 1,2
%A A331809 _Zhandos Mambetaliyev_, Jan 27 2020
%E A331809 a(12)-a(15) from _Hugo Pfoertner_, Jan 27 2020
%E A331809 More terms, using _Rémy Sigrist_'s C++ at A331811 from _Hugo Pfoertner_, Jan 28 2020