A365381 -id:A365381 - cp's OEIS frontend

A365544 Number of subsets of {1..n} containing two distinct elements summing to n.

0, 0, 0, 2, 4, 14, 28, 74, 148, 350, 700, 1562, 3124, 6734, 13468, 28394, 56788, 117950, 235900, 484922, 969844, 1979054, 3958108, 8034314, 16068628, 32491550, 64983100, 131029082, 262058164, 527304974, 1054609948, 2118785834, 4237571668, 8503841150, 17007682300

Offset: 0

Gus Wiseman, Sep 20 2023

			The a(1) = 0 through a(5) = 14 subsets:
  .  .  {1,2}    {1,3}      {1,4}
        {1,2,3}  {1,2,3}    {2,3}
                 {1,3,4}    {1,2,3}
                 {1,2,3,4}  {1,2,4}
                            {1,3,4}
                            {1,4,5}
                            {2,3,4}
                            {2,3,5}
                            {1,2,3,4}
                            {1,2,3,5}
                            {1,2,4,5}
                            {1,3,4,5}
                            {2,3,4,5}
                            {1,2,3,4,5}

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

For strict partitions we have A140106 shifted left.
The version for partitions is A004526.
The complement is counted by A068911.
For all subsets of elements we have A365376.
Main diagonal k = n of A365541.
A000009 counts subsets summing to n.
A007865/A085489/A151897 count certain types of sum-free subsets.
A093971/A088809/A364534 count certain types of sum-full subsets.
A365381 counts subsets with a subset summing to k.
Cf. A008967, A095944, A167762, A238628, A288728, A326083, A364272, A365377.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n]&]],{n,0,10}]

def A365544(n): return (1<>1)<<1 if n&1 else 3**(n-1>>1)<<2) if n else 0 # Chai Wah Wu, Aug 30 2024

a(n) = 2^n - A068911(n).
From Alois P. Heinz, Aug 30 2024: (Start)
G.f.: 2*x^3/((2*x-1)*(3*x^2-1)).
a(n) = 2 * A167762(n-1) for n>=1. (End)

A365322 Number of subsets of {1..n} that cannot be linearly combined using positive coefficients to obtain n.

Original entry on oeis.org

0, 1, 2, 5, 11, 26, 54, 116, 238, 490, 994, 2011, 4045, 8131, 16305, 32672, 65412, 130924, 261958, 524066, 1048301, 2096826, 4193904, 8388135, 16776641, 33553759, 67108053, 134216782, 268434324, 536869595, 1073740266, 2147481835, 4294965158, 8589932129

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

			The set {1,3} has 4 = 1 + 3 so is not counted under a(4). However, 3 cannot be written as a linear combination of {1,3} using all positive coefficients, so it is counted under a(3).
The a(1) = 1 through a(4) = 11 subsets:
  {}  {}     {}       {}
      {1,2}  {2}      {3}
             {1,3}    {1,4}
             {2,3}    {2,3}
             {1,2,3}  {2,4}
                      {3,4}
                      {1,2,3}
                      {1,2,4}
                      {1,3,4}
                      {2,3,4}
                      {1,2,3,4}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement is counted by A088314.
The version for strict partitions is A088528.
The nonnegative complement is counted by A365073, without n A365542.
The binary complement is A365315, nonnegative A365314.
The binary version is A365321, nonnegative A365320.
For nonnegative coefficients we have A365380.
A085489 and A364755 count subsets without the sum of two distinct elements.
A124506 appears to count combination-free subsets, differences of A326083.
A179822 counts sum-closed subsets, first differences of A326080.
A364350 counts combination-free strict partitions, non-strict A364915.
A365046 counts combination-full subsets, first differences of A364914.
Cf. A070880, A088809, A093971, A151897, A326020, A364272, A364839, A365043, A365381.

b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[], i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> 2^n-nops(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Sep 04 2023

cpu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],cpu[n,#]=={}&]],{n,0,10}]

from sympy.utilities.iterables import partitions
def A365322(n): return (1<Chai Wah Wu, Sep 14 2023

a(n) = 2^n - A088314(n).

a(n) = A070880(n) + 2^(n-1) for n>=1.

More terms from Alois P. Heinz, Sep 04 2023

A367404 Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 5, 3, 4, 2, 3, 7, 5, 6, 4, 3, 3, 11, 7, 9, 6, 6, 3, 4, 15, 11, 13, 10, 9, 6, 4, 4, 22, 15, 20, 13, 15, 9, 8, 4, 5, 30, 22, 27, 21, 21, 15, 12, 8, 5, 5, 42, 30, 39, 28, 30, 21, 20, 12, 10, 5, 6, 56, 42, 53, 41, 42, 33, 28, 20, 15, 10, 6, 6

Offset: 2

Gus Wiseman, Nov 17 2023

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (3,2,1,1) has semi-sum 3 = 2+1, but no semi-sum 6, so y is counted under T(7,3) but not under T(7,6).
Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   2
   5   3   4   2   3
   7   5   6   4   3   3
  11   7   9   6   6   3   4
  15  11  13  10   9   6   4   4
  22  15  20  13  15   9   8   4   5
  30  22  27  21  21  15  12   8   5   5
  42  30  39  28  30  21  20  12  10   5   6
  56  42  53  41  42  33  28  20  15  10   6   6
  77  56  73  55  60  42  44  28  25  15  12   6   7
Row n = 7 counts the following partitions:
  (511)      (421)     (331)    (421)   (511)  (61)
  (4111)     (3211)    (322)    (4111)  (421)  (52)
  (3211)     (2221)    (3211)   (322)   (331)  (43)
  (31111)    (22111)   (31111)  (3211)
  (22111)    (211111)  (2221)
  (211111)             (22111)
  (1111111)

Column k = 0 is A000041.
Column n = k is A004526.
The complement for all submultisets is A046663, strict A365663.
For subsets instead of partitions we have A365541, non-binary A365381.
The non-binary version is A365543, strict A365661.
Row sums are A366738.
The strict case is A367405.
Cf. A122768, A108917, A299701, A304792, A364272, A364911, A365658.

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#, {2}],k]&]], {n,2,10}, {k,2,n}]

A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 1, 0, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 2, 4, 3, 4, 4, 5, 3, 4, 5, 3, 3, 5, 4, 4, 5, 5, 5, 4, 4, 6, 4, 3, 6, 5, 6, 5, 7, 5, 7, 4, 5, 6, 5, 5, 7, 7, 8, 7, 8, 8, 7, 7, 5, 5, 7

Offset: 3

Gus Wiseman, Nov 18 2023

			Triangle begins:
  1
  0  1
  0  0  2
  1  1  1  2
  1  0  1  1  3
  1  1  1  1  2  3
  1  1  1  2  2  2  4
  2  2  3  2  3  2  3  4
  2  2  3  2  3  3  3  3  5
  3  2  4  3  4  4  5  3  4  5
  3  3  5  4  4  5  5  5  4  4  6
  4  3  6  5  6  5  7  5  7  4  5  6
  5  5  7  7  8  7  8  8  7  7  5  5  7
  6  5  9  8 10  7 10  9 10  7  9  5  6  7
  7  7 10 10 12 11 11 11 12 10  9  9  6  6  8
  9  7 13 11 15 12 13 13 15 13 13  9 11  6  7  8
Row n = 9 counts the following strict partitions:
  (6,2,1)  (5,3,1)  (4,3,2)  (5,3,1)  (6,2,1)  (6,2,1)  (8,1)
                             (4,3,2)  (4,3,2)  (5,3,1)  (7,2)
                                                        (6,3)
                                                        (5,4)
Row n = 13 counts the following strict partitions (A=10, B=11, C=12):
  A21   931   841   751   652   751   841   931   A21  A21  C1
  7321  7321  832   742   643   7321  742   832   832  931  B2
  6421  5431  7321  6421  6421  652   7321  7321  742  841  A3
              6421  5431  5431  6421  643   643   652  751  94
              5431              5431  5431  6421            85
                                                            76

Column n = k is A004526.
Column k = 3 is A025148.
For subsets instead of partitions we have A365541, non-binary A365381.
The non-binary version is A365661, non-strict A365543.
The non-binary complement is A365663, non-strict A046663.
Row sums are A366741, non-strict A366738.
The non-strict version is A367404.
Cf. A000041, A088809, A093971, A122768, A108917, A284640, A304792, A364272, A364911, A365658.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], k]&]], {n,3,10}, {k,3,n}]

A367396 Number of subsets of {1..n} whose cardinality is the sum of two distinct elements.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 17, 40, 90, 199, 435, 939, 2007, 4258, 8976, 18817, 39263, 81595, 168969, 348820, 718134, 1474863, 3022407, 6181687, 12621135, 25727686, 52369508, 106460521, 216162987, 438431215, 888359841, 1798371648, 3637518354, 7351824439, 14848255803

Offset: 0

Gus Wiseman, Nov 21 2023

nonn

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is not counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                    {1,2,4}    {1,2,4}      {1,2,4}
                    {1,2,3,4}  {1,2,5}      {1,2,5}
                               {1,2,3,4}    {1,2,6}
                               {1,2,3,5}    {1,2,3,4}
                               {1,3,4,5}    {1,2,3,5}
                               {1,2,3,4,5}  {1,2,3,6}
                                            {1,3,4,5}
                                            {1,3,4,6}
                                            {1,3,5,6}
                                            {1,2,3,4,5}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396*  A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A002865 counts partitions whose length is a part, complement A229816.
A364534 counts sum-full subsets.
A088809 and A093971 count subsets containing semi-sums.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A236912, A237113, A237667, A237668, A304792, A363225, A364272, A365658, A366131, A366740.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],Length[#]]&]],{n,0,10}]

from itertools import combinations
def A367396(n): return sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - 5*a(n-4) + 2*a(n-5) for n > 4.
G.f.: x^3*(x - 1)/((2*x - 1)*(x^4 - 2*x^3 + x^2 - 2*x + 1)). (End)

a(18)-a(33) from Chai Wah Wu, Nov 21 2023

a(34) from Paul Muljadi, Nov 24 2023

A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

Original entry on oeis.org

4, 12, 18, 30, 36, 40, 42, 54, 60, 66, 78, 81, 90, 100, 102, 112, 114, 120, 126, 135, 138, 140, 150, 168, 174, 180, 186, 189, 198, 210, 220, 222, 225, 234, 246, 250, 252, 258, 260, 270, 280, 282, 297, 300, 306, 315, 318, 330, 336, 340, 342, 350, 351, 352, 354

Offset: 1

Gus Wiseman, Nov 21 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

These are the Heinz numbers of the partitions counted by A367394.

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397*  A367401
A325761 ranks partitions whose length is a part, counted by A002865.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A304792 counts subset-sums of partitions, strict A365925.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A229816, A237667, A237668, A238628, A363225, A364272, A365543, A365658, A365918, A366740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MemberQ[Total/@Subsets[prix[#],{2}],PrimeOmega[#]]&]

A367400 Number of subsets of {1..n} whose cardinality is not the sum of two distinct elements.

Original entry on oeis.org

1, 2, 4, 7, 13, 25, 47, 88, 166, 313, 589, 1109, 2089, 3934, 7408, 13951, 26273, 49477, 93175, 175468, 330442, 622289, 1171897, 2206921, 4156081, 7826746, 14739356, 27757207, 52272469, 98439697, 185381983, 349112000, 657448942, 1238110153

Offset: 0

Gus Wiseman, Nov 21 2023

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is counted under a(8).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,3,4}
                         {2,3,4}

The version containing n appears to be A112575.
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400*
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A002865 counts partitions whose length is a part, complement A229816.
A364534 counts sum-full subsets.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A237113, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366131, A366740.

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n,0,10}]

from itertools import combinations
def A367400(n): return (n*(n+1)>>1)+1+sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if not any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) for n > 3.
G.f.: (-x^3 + x^2 + 1)/(x^4 - 2*x^3 + x^2 - 2*x + 1). (End)

a(18)-a(33) from Chai Wah Wu, Nov 21 2023

A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Nov 21 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

These are the Heinz numbers of the partitions counted by A367398.

			60 has semiprime divisor 10 with prime indices {1,3} summing to 4 = bigomega(60), so 60 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401*
A002865 counts partitions w/ length, complement A229816, ranks A325761.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366740.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], FreeQ[Total/@Subsets[prix[#],{2}], PrimeOmega[#]]&]

A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 2, 5, 3, 2, 3, 1, 3, 2, 3, 6, 3, 3, 4, 3, 3, 4, 3, 3, 8, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 7, 6, 7, 7, 7, 4, 7, 7, 7, 6, 7, 15, 8, 7, 8, 8, 8, 8, 8

Offset: 1

Clark Kimberling, Feb 05 2014

			First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions:  [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000

Column k = n is A000009.
Column k = 2 is A015744.
Column k = 1 is A025147.
The non-strict complement is obtained by adding zeros after A046663.
Diagonal n = 2k is A237258.
Row sums are A284640.
For subsets instead of partitions we have A365381.
The non-strict version is obtained by removing column k = 0 from A365543.
Including column k = 0 gives A365661.
The complement is obtained by adding zeros after A365663.
Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.

Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%]  (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* Gus Wiseman, Nov 16 2023 *)

T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.

A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

Original entry on oeis.org

0, 0, 2, 2, 10, 14, 46, 74, 202, 350, 862, 1562, 3610, 6734, 14926, 28394, 61162, 117950, 249022, 484922, 1009210, 1979054, 4076206, 8034314, 16422922, 32491550, 66045982, 131029082, 265246810, 527304974, 1064175886, 2118785834, 4266269482, 8503841150, 17093775742, 34101458042, 68461196410, 136664112494

Offset: 0

Gus Wiseman, Oct 07 2023

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1}    {1,2}    {2}        {1,4}
        {1,2}  {1,2,3}  {1,2}      {2,3}
                        {1,3}      {1,2,3}
                        {2,3}      {1,2,4}
                        {2,4}      {1,3,4}
                        {1,2,3}    {1,4,5}
                        {1,2,4}    {2,3,4}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {1,2,3,4}
                        {1,2,3,4}  {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

The complement is counted by A117855.
For pairs summing to n + 1 we have A167936.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A008967, A167762, A238628, A365376, A365377, A365381, A365541, A365544.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}]

def A366131(n): return (1<>1)<<1) if n else 0 # Chai Wah Wu, Nov 14 2023

From Chai Wah Wu, Nov 14 2023: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) for n > 3.
G.f.: 2*x^2*(1 - x)/((2*x - 1)*(3*x^2 - 1)). (End)

cp's OEIS Frontend

A365544 Number of subsets of {1..n} containing two distinct elements summing to n.

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A365322 Number of subsets of {1..n} that cannot be linearly combined using positive coefficients to obtain n.

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A367404 Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.

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A367405 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.

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A367396 Number of subsets of {1..n} whose cardinality is the sum of two distinct elements.

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A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

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A367400 Number of subsets of {1..n} whose cardinality is not the sum of two distinct elements.

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A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

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