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A288888 a(n) is the total number of elements in all sum-free subsets of {1,...,n}.

1, 2, 7, 12, 27, 45, 93, 144, 294, 428, 796, 1220, 2186, 3155, 5637, 8102, 13907, 20070, 33746, 47416, 81050, 112226, 184541, 260780, 421222, 577447, 947934, 1304821, 2087701, 2857024, 4535223, 6157288, 9878133, 13257735, 20790674, 28332734, 44304037, 59072318

Offset: 1

Ben Burns, Jun 18 2017

nonn

a(n) is the sum of the n-th row of A288887.

			A288887 begins:
  1;
  1,  1;
  2,  2,  3;
  3,  2,  4,  3;
  5,  3,  7,  5,  7;
  ...

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..76
Eric Weisstein's World of Mathematics, Sum-Free Set

Cf. A007865. Rows sums of triangle A288887.

sumfree(v) = {for(i=1, #v, for (j=1, i, if (setsearch(v, v[i]+v[j]), return (0)););); return (1);}
a(n) = {my(nb = 0); forsubset(n, s, if (#s && sumfree(Set(s)), nb += #s);); nb;} \\ Michel Marcus, Nov 08 2020

A288728 Number of sum-free sets that can be created by adding n to all sum-free sets [1..n-1].

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 18, 19, 47, 43, 102, 116, 238, 240, 553, 554, 1185, 1259, 2578, 2607, 5873, 5526, 11834, 12601, 24692, 24390, 53735, 52534, 107445, 107330, 218727, 215607, 461367, 427778, 891039, 910294, 1804606, 1706828, 3695418, 3411513, 7136850, 6892950

Offset: 1

Ben Burns, Jun 14 2017

nonn

Using the standard definition of sum-free set, this is simply the difference of successive terms in A007865.

Number of subsets of {1..n} containing n but not containing the sum of any other two elements (repeats allowed). Also the number of sum-free sets (A007865) with maximum n. - Gus Wiseman, Aug 12 2023

			1 can be added to {};
2 can be added to {} but not {1};
3 can be added to {},{1},{2};
4 can be added to {},{1},{3} but not {2},{1,3},{2,3}.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(1) = 1 through a(7) = 18 sum-free sets with maximum n:
  {1}  {2}  {3}    {4}    {5}      {6}      {7}
            {1,3}  {1,4}  {1,5}    {1,6}    {1,7}
            {2,3}  {3,4}  {2,5}    {2,6}    {2,7}
                          {3,5}    {4,6}    {3,7}
                          {4,5}    {5,6}    {4,7}
                          {1,3,5}  {1,4,6}  {5,7}
                          {3,4,5}  {2,5,6}  {6,7}
                                   {4,5,6}  {1,3,7}
                                            {1,4,7}
                                            {1,5,7}
                                            {2,3,7}
                                            {2,6,7}
                                            {3,5,7}
                                            {4,5,7}
                                            {4,6,7}
                                            {5,6,7}
                                            {1,3,5,7}
                                            {4,5,6,7}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..88
Eric Weisstein's World of Mathematics, Sum-Free Set

Cf. A007865.
For non-binary sum-free subsets of {1..n} we have A237667.
For sum-free partitions we have A364345, without re-using parts A236912.
Without re-using parts we have A364755, diffs of A085489 (non-bin A151897).
The complement without re-using parts is A364756, differences of A088809.
Cf. A002865, A093971, A237113, A237668, A326083, A364347, A364348, A364534.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Intersection[#,Total/@Tuples[#,2]]=={}&]],{n,10}] (* Gus Wiseman, Aug 12 2023 *)

a(n) = A007865(n) - A007865(n-1).

A288887 Triangle read by rows: T(n,k) is the number of times k is a member of a sum-free subset of {1, ..., n} for 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 2, 4, 3, 5, 3, 7, 5, 7, 7, 5, 7, 8, 10, 8, 12, 8, 12, 13, 17, 13, 18, 16, 13, 17, 13, 23, 18, 25, 19, 28, 21, 30, 21, 40, 32, 43, 32, 47, 37, 29, 40, 29, 40, 45, 57, 45, 63, 43, 62, 44, 66, 45, 65, 72, 95, 71, 104, 70, 102, 85, 66, 95, 71, 89, 72, 132, 109, 139, 104, 142, 116

Offset: 1

Ben Burns, Jun 18 2017

			For the nine sum-free subsets of {1,2,3,4}, 1 is a member of three of them, 2 is a member of two, 3 is a member of four, and 4 is a member of three, hence the 4th row is 3,2,4,3.
The triangle begins:
   1;
   1,  1;
   2,  2,  3;
   3,  2,  4,  3;
   5,  3,  7,  5,  7;
   7,  5,  7,  8, 10,  8;
  12,  8, 12, 13, 17, 13, 18;
  16, 13, 17, 13, 23, 18, 25, 19;
  ...

Fausto A. C. Cariboni, Rows n = 1..70, flattened
Ben Burns, C# program to generate a(n) up to n=64
Eric Weisstein's World of Mathematics, Sum-Free Set

Cf. A007865, A288888 (row sums).

sumfree(v) = {for(i=1, #v, for (j=1, i, if (setsearch(v, v[i]+v[j]), return (0)););); return (1);}
row(n) = {my(v = vector(n)); forsubset(n, s, if (sumfree(Set(s)), for (k=1, n, if (setsearch(Set(s), k), v[k]++);););); v;} \\ Michel Marcus, Nov 08 2020

A288817 Number of set partitions of [n] such that each subset is sum-free.

Original entry on oeis.org

1, 1, 1, 3, 6, 20, 67, 291, 1099, 5780, 26249, 153238, 832366, 5443440, 32738738, 239515824, 1591963864, 12548347149, 93066370414

Offset: 0

Ben Burns, Jun 17 2017

The count can be built constructively by listing all possible sum-free sets of partitions into collections containing {1, ..., n}. For n>1, iterate over the previous generation and insert n into each partition if the result is sum-free, and also append n to the end as its own partition. See Example.

			Where "|" is a partition divider, then:
a(1)=1, given by { 1 }.
a(2)=1, given by { 1|2 }.
a(3)=3, given by { 1,3|2 }, { 1|2,3 }, { 1|2|3 }.
a(4)=6, given by { 1,3|2|4 }, { 1,4|2,3 }, { 1|2,3|4 }, { 1,4|2|3 }, { 1|2|3,4 }, { 1|2|3|4 }.

Ben Burns, C# program to generate counts
Eric Weisstein's World of Mathematics, Sum-Free Set

Cf. A007865, A000041.

A245390 Number of ways to build an expression using non-redundant parentheses in an expression of n variables, counted in decreasing sub-partitions of equal size.

Original entry on oeis.org

1, 1, 3, 11, 39, 145, 514, 1901, 6741, 24880, 87791, 325634, 1152725, 4251234, 15052387, 55750323, 197031808, 729360141, 2579285955, 9539017676, 33822222227, 124889799518, 440743675148, 1635528366655, 5790967247863, 21360573026444, 75643815954959, 280096917425535

Offset: 1

Ben Burns, Jul 22 2014

nonn

Expressions contain only binary operators. Parentheses around a single variable or the entire expression are considered redundant or unnecessary.

If the number of partitions (parenthetical group) does not evenly divide the number of variables, combinations of remaining variables are counted but not divided into sub-partitions.

			For n=3, the a(3)=3 different possibilities are
1)  x+x+x,
2)  (x+x)+x,
3)  x+(x+x).
For n=4, the a(4)=11 different possibilities are
1)  x+x+x+x,
2)  (x+x+x)+x,
3)  ((x+x)+x)+x,
4)  (x+(x+x))+x,
5)  x+(x+x+x),
6)  x+((x+x)+x),
7)  x+(x+(x+x)),
8)  (x+x)+x+x,
9)  x+(x+x)+x,
10) x+x+(x+x),
11) (x+x)+(x+x).
For n=5, a(5)=39 is the first time this sequence differs from A001003. The combinations for (x+x+x)+(x+x) and (x+x)+(x+x+x) are not counted as these two partitions are not of equal size.

Ben Burns, Table of n, a(n) for n = 1..98

Cf. A001003

from functools import reduce
import math
import operator as op
def binom(n, r):
    r = min(r, n-r)
    if r == 0: return 1
    numer = reduce(op.mul, range(n, n-r, -1))
    denom = reduce(op.mul, range(1, r+1))
    return numer//denom
max = 100
seq = [0,1,1]
for n in range(3, max) :
    sum = 0
    for choose in range(2, n+1):
        f = math.ceil(n/choose)
        f+=1
        for times in range(1, int(f)) :
            if choose*times > n :
                continue
            if choose==n and times==1 :
                sum += 1
            else :
                sum += binom(n - (choose*times) + times, times) * (seq[choose]**times)
    seq.append(sum)
for (i, x) in enumerate(seq) :
    if i==0:
        continue
    print(i, x)

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User: Ben Burns

A288888 a(n) is the total number of elements in all sum-free subsets of {1,...,n}.

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A288728 Number of sum-free sets that can be created by adding n to all sum-free sets [1..n-1].

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A288887 Triangle read by rows: T(n,k) is the number of times k is a member of a sum-free subset of {1, ..., n} for 1 <= k <= n.

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A288817 Number of set partitions of [n] such that each subset is sum-free.

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A245390 Number of ways to build an expression using non-redundant parentheses in an expression of n variables, counted in decreasing sub-partitions of equal size.

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Python