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A051912 a(n) is the smallest integer such that the sum of any three ordered terms a(k), k <= n, is unique.

%I A051912 #36 Mar 01 2024 14:15:45
%S A051912 0,1,4,13,32,71,124,218,375,572,744,1208,1556,2441,3097,4047,5297,
%T A051912 6703,7838,10986,12331,15464,19143,24545,28973,34405,37768,45863,
%U A051912 50876,61371,68302,77917,88544,101916,122031,131624,148574,171236,197814
%N A051912 a(n) is the smallest integer such that the sum of any three ordered terms a(k), k <= n, is unique.
%H A051912 Chai Wah Wu, <a href="/A051912/b051912.txt">Table of n, a(n) for n = 0..225</a> (terms 0..100 from Robert Israel)
%e A051912 Three terms chosen from {0,1,4} can be 0+0+0; 0+0+1; 0+1+1; 1+1+1; 0+0+4; 0+1+4; 1+1+4; 0+4+4; 1+4+4; 4+4+4 are all distinct (3*4*5/6 = 10 terms), so a(2) = 4 is the next integer of the sequence after 0 and 1.
%p A051912 A[0]:= 0: S:= {0}: S2:= {0}: S3:= {0}:
%p A051912 for i from 1 to 40 do
%p A051912   for x from A[i-1] do
%p A051912     if (map(t -> t+x,S2) intersect S3 = {}) and (map(t -> t+2*x, S) intersect S3 = {}) then
%p A051912      A[i]:= x;
%p A051912      S3:= S3 union map(t -> t+x,S2) union map(t -> t+2*x, S) union {3*x};
%p A051912      S2:= S2 union map(t -> t+x, S) union {2*x};
%p A051912      S:= S union {x};
%p A051912      break
%p A051912     fi
%p A051912   od
%p A051912 od:
%p A051912 seq(A[i],i=0..40); # _Robert Israel_, Jul 01 2019
%t A051912 a[0] = 0; a[1] = 1; a[n_] := a[n] = For[A0 = Array[a, n, 0]; an = a[n-1] + 1, True, an++, A1 = Append[A0, an]; A2 = Flatten[Table[A1[[{i, j, k}]], {i, 1, n+1}, {j, i, n+1}, {k, j, n+1}], 2]; A3 = Sort[Total /@ A2]; If[Length[A3] == Length[Union[A3]], Return[an]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 38}] (* _Jean-François Alcover_, Nov 24 2016 *)
%o A051912 (Python)
%o A051912 from itertools import count, islice
%o A051912 def A051912_gen(): # generator of terms
%o A051912     aset1, aset2, aset3, alist = set(), set(), set(), []
%o A051912     for k in count(0):
%o A051912         bset2, bset3 = {k<<1}, {3*k}
%o A051912         if 3*k not in aset3:
%o A051912             for d in aset1:
%o A051912                 if (m:=d+(k<<1)) in aset3:
%o A051912                     break
%o A051912                 bset2.add(d+k)
%o A051912                 bset3.add(m)
%o A051912             else:
%o A051912                 for d in aset2:
%o A051912                     if (m:=d+k) in aset3:
%o A051912                         break
%o A051912                     bset3.add(m)
%o A051912                 else:
%o A051912                     yield k
%o A051912                     alist.append(k)
%o A051912                     aset1.add(k)
%o A051912                     aset2 |= bset2
%o A051912                     aset3 |= bset3
%o A051912 A051912_list = list(islice(A051912_gen(),20)) # _Chai Wah Wu_, Sep 01 2023
%Y A051912 Row 3 of A365515.
%Y A051912 Cf. A025582, A036241, A062065.
%K A051912 nonn,nice
%O A051912 0,3
%A A051912 _Wouter Meeussen_, Dec 17 1999
%E A051912 More terms from _Naohiro Nomoto_, Jul 22 2001