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A010672 A B_2 sequence: a(n) = least value such that the sequence increases and pairwise sums of distinct terms are all distinct.

%I A010672 #39 Jan 14 2025 23:10:22
%S A010672 0,1,2,4,7,12,20,29,38,52,73,94,127,151,181,211,257,315,373,412,475,
%T A010672 530,545,607,716,797,861,964,1059,1160,1306,1385,1434,1555,1721,1833,
%U A010672 1933,2057,2260,2496,2698,2873,3060,3196,3331,3628,3711,3867,4139,4446,4639
%N A010672 A B_2 sequence: a(n) = least value such that the sequence increases and pairwise sums of distinct terms are all distinct.
%H A010672 Giovanni Resta, <a href="/A010672/b010672.txt">Table of n, a(n) for n = 0..10000</a> (first 7611 terms from D. Mondot)
%H A010672 <a href="/index/Br#B_2">Index entries for B_2 sequences</a>.
%F A010672 a(n) = A011185(n+1) - 1. - _Robert Israel_, May 02 2016
%p A010672 N:= 10^6: # to get all terms <= N
%p A010672 A[0]:= 0: Delta:= {}: As:= {A[0]}:
%p A010672 Cands:= {$1..N}:
%p A010672 for n from 1  while Cands <> {} do
%p A010672   A[n]:= min(Cands);
%p A010672   Cands:= Cands minus ({A[n]} union map(`+`,Delta, A[n]));
%p A010672   Delta:= Delta union map(t ->A[n] - t, As);
%p A010672   As:= As union {A[n]};
%p A010672 od:
%p A010672 seq(A[i],i=0..n-1); # _Robert Israel_, May 02 2016
%t A010672 seq = {0};
%t A010672 pairSums = {};
%t A010672 nextTerm = 1;
%t A010672 pairSumsUpdate := Join[pairSums, nextTerm + seq]
%t A010672 hasDuplicates := ! DuplicateFreeQ[pairSumsUpdate]
%t A010672 Do[
%t A010672  While[hasDuplicates, nextTerm++];
%t A010672  pairSums = pairSumsUpdate;
%t A010672  AppendTo[seq, nextTerm];
%t A010672  , 50]
%t A010672 seq (* _David Trimas_, Dec 28 2024 *)
%o A010672 (MATLAB)
%o A010672 N = 3*10^8; % to get all terms < N
%o A010672 Cands = ones(N,1);
%o A010672 Delta = [];
%o A010672 A = [];
%o A010672 n = 1;
%o A010672 while nnz(Cands) > 0
%o A010672       A(n) = find(Cands,1,'first');
%o A010672       Cands(A(n)) = 0;
%o A010672       Rem = Delta(Delta <= N - A(n)) + A(n);
%o A010672       Cands(Rem) = 0;
%o A010672       Delta = union(Delta, -A(1:n-1)+A(n));
%o A010672       if mod(n,10)==0
%o A010672        fprintf('a(%d)=%d\n',n,A(n));
%o A010672        toc;
%o A010672       end
%o A010672       n = n + 1;
%o A010672 end
%o A010672 A - 1 % _Robert Israel_, May 02 2016
%o A010672 (Python)
%o A010672 from itertools import count, islice
%o A010672 def A010672_gen(): # generator of terms
%o A010672     aset2, alist = set(), []
%o A010672     for k in count(0):
%o A010672         bset2 = set()
%o A010672         for a in alist:
%o A010672             if (b:=a+k) in aset2:
%o A010672                 break
%o A010672             bset2.add(b)
%o A010672         else:
%o A010672             yield k
%o A010672             alist.append(k)
%o A010672             aset2.update(bset2)
%o A010672 A010672_list = list(islice(A010672_gen(),30)) # _Chai Wah Wu_, Sep 11 2023
%Y A010672 A025582 is a similar sequence, but there the pairwise sums of (not necessarily distinct) elements are all distinct.
%Y A010672 Cf. A011185.
%K A010672 nonn
%O A010672 0,3
%A A010672 _Dan Hoey_