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

0, 1, 2, 4, 7, 12, 20, 29, 38, 52, 73, 94, 127, 151, 181, 211, 257, 315, 373, 412, 475, 530, 545, 607, 716, 797, 861, 964, 1059, 1160, 1306, 1385, 1434, 1555, 1721, 1833, 1933, 2057, 2260, 2496, 2698, 2873, 3060, 3196, 3331, 3628, 3711, 3867, 4139, 4446, 4639

Offset: 0

Author

Dan Hoey

Keywords

• nonn

Links

• Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 7611 terms from D. Mondot)
• Index entries for B_2 sequences.

Crossrefs

A025582 is a similar sequence, but there the pairwise sums of (not necessarily distinct) elements are all distinct.

Cf. A011185.

Programs

MATLAB

N = 3*10^8; % to get all terms < N
Cands = ones(N,1);
Delta = [];
A = [];
n = 1;
while nnz(Cands) > 0
      A(n) = find(Cands,1,'first');
      Cands(A(n)) = 0;
      Rem = Delta(Delta <= N - A(n)) + A(n);
      Cands(Rem) = 0;
      Delta = union(Delta, -A(1:n-1)+A(n));
      if mod(n,10)==0
       fprintf('a(%d)=%d\n',n,A(n));
       toc;
      end
      n = n + 1;
end
A - 1 % Robert Israel, May 02 2016

Maple

N:= 10^6: # to get all terms <= N
A[0]:= 0: Delta:= {}: As:= {A[0]}:
Cands:= {$1..N}:
for n from 1  while Cands <> {} do
  A[n]:= min(Cands);
  Cands:= Cands minus ({A[n]} union map(`+`,Delta, A[n]));
  Delta:= Delta union map(t ->A[n] - t, As);
  As:= As union {A[n]};
od:
seq(A[i],i=0..n-1); # Robert Israel, May 02 2016

Mathematica

seq = {0};
pairSums = {};
nextTerm = 1;
pairSumsUpdate := Join[pairSums, nextTerm + seq]
hasDuplicates := ! DuplicateFreeQ[pairSumsUpdate]
Do[
 While[hasDuplicates, nextTerm++];
 pairSums = pairSumsUpdate;
 AppendTo[seq, nextTerm];
 , 50]
seq (* David Trimas, Dec 28 2024 *)

Python

from itertools import count, islice
def A010672_gen(): # generator of terms
    aset2, alist = set(), []
    for k in count(0):
        bset2 = set()
        for a in alist:
            if (b:=a+k) in aset2:
                break
            bset2.add(b)
        else:
            yield k
            alist.append(k)
            aset2.update(bset2)
A010672_list = list(islice(A010672_gen(),30)) # Chai Wah Wu, Sep 11 2023

Formula

a(n) = A011185(n+1) - 1. - Robert Israel, May 02 2016