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A247184 a(0) = 0. a(n) is the number of distinct sums of two elements in [a(0), ... a(n-1)] chosen without replacement. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 28 12:59 EDT 2022. Contains 354907 sequences. (Running on oeis4.)