A247184 a(0) = 0. a(n) is the number of distinct sums of two elements in [a(0), ... a(n-1)] chosen without replacement.

0, 0, 1, 2, 4, 7, 11, 15, 20, 26, 32, 40, 48, 57, 65, 73, 81, 90, 98, 106, 114, 123, 132, 147, 157, 170, 190, 202, 223, 236, 251, 270, 291, 314, 338, 361, 380, 398, 421, 443, 471, 495, 520, 544, 567, 592, 616, 639, 663, 692, 720, 749, 781, 819, 852, 885, 913, 948, 987, 1023, 1055, 1088

Offset

0,4

Comments

a(n) <= A000217(n)-n.

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.

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.

a(i)+a(j) and a(j)+a(i) are regarded as the same for all indices i and j.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Example

a(1) gives the number of distinct sums of two elements of [0]. There aren't two elements so a(1) = 0.
a(2) gives the number of distinct sums of two elements of [0,0]. There is only 1 sum, 0, so a(2) = 1.
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.
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.

Maple

s:= proc(n) option remember; `if`(n=0, {},
      {s(n-1)[], seq(a(i)+a(n), i=0..n-1)})
    end:
a:= proc(n) option remember;
      `if`(n=0, 0, nops(s(n-1)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 16 2020

Mathematica

s[n_] := s[n] = If[n == 0, {},
   Union@Join[s[n-1], Table[a[i] + a[n], {i, 0, n-1}]]];
a[n_] := a[n] =
   If[n == 0, 0, Length[s[n-1]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 16 2021, after Alois P. Heinz *)

Prog

(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

Crossrefs

Cf. A000217, A008585, A247185.

Sequence in context: A198759 A078617 A199085 * A025703 A025709 A036700

Adjacent sequences: A247181 A247182 A247183 * A247185 A247186 A247187

Keyword

nonn

Author

Derek Orr, Nov 22 2014

Status

approved