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%I #43 Jul 16 2021 07:51:19
%S 0,0,1,2,4,7,11,15,20,26,32,40,48,57,65,73,81,90,98,106,114,123,132,
%T 147,157,170,190,202,223,236,251,270,291,314,338,361,380,398,421,443,
%U 471,495,520,544,567,592,616,639,663,692,720,749,781,819,852,885,913,948,987,1023,1055,1088
%N a(0) = 0. a(n) is the number of distinct sums of two elements in [a(0), ... a(n-1)] chosen without replacement.
%C a(n) <= A000217(n)-n.
%C Without replacement means that a(i)+a(i) is not a valid sum to include. However, if a(i) = a(j), a(i)+a(j) is still a valid sum to include because they have different indices.
%C If you include a(i)+a(i) (i.e., with replacement) as a valid sum, the sequence becomes 0, 1, 3, 6, 9, 12, ... = 0, 1, followed by A008585(n) for n > 0.
%C a(i)+a(j) and a(j)+a(i) are regarded as the same for all indices i and j.
%H Alois P. Heinz, <a href="/A247184/b247184.txt">Table of n, a(n) for n = 0..10000</a>
%e a(1) gives the number of distinct sums of two elements of [0]. There aren't two elements so a(1) = 0.
%e a(2) gives the number of distinct sums of two elements of [0,0]. There is only 1 sum, 0, so a(2) = 1.
%e a(3) gives the number of distinct sums of two elements of [0,0,1]. There are 2 distinct possible sums 0 and 1, so a(3) = 2.
%e a(4) gives the number of distinct sums of two elements of [0,0,1,2]. There are 4 distinct possible sums {0, 1, 2, 3}, so a(4) = 4.
%p s:= proc(n) option remember; `if`(n=0, {},
%p {s(n-1)[], seq(a(i)+a(n), i=0..n-1)})
%p end:
%p a:= proc(n) option remember;
%p `if`(n=0, 0, nops(s(n-1)))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Nov 16 2020
%t s[n_] := s[n] = If[n == 0, {},
%t Union@Join[s[n-1], Table[a[i] + a[n], {i, 0, n-1}]]];
%t a[n_] := a[n] =
%t If[n == 0, 0, Length[s[n-1]]];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jul 16 2021, after _Alois P. Heinz_ *)
%o (PARI) v=[0];n=1;while(n<75,w=[];for(i=1,#v,for(j=i+1,#v,w=concat(w,v[i]+v[j])));v=concat(v,#vecsort(w,,8));n++);v
%Y Cf. A000217, A008585, A247185.
%K nonn
%O 0,4
%A _Derek Orr_, Nov 22 2014
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