A381476 -id:A381476 - cp's OEIS frontend

A143823 Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.

1, 2, 4, 7, 13, 22, 36, 57, 91, 140, 216, 317, 463, 668, 962, 1359, 1919, 2666, 3694, 5035, 6845, 9188, 12366, 16417, 21787, 28708, 37722, 49083, 63921, 82640, 106722, 136675, 174895, 222558, 283108, 357727, 451575, 567536, 712856, 890405, 1112081, 1382416, 1717540

Offset: 0

John W. Layman, Sep 02 2008

When the set {x(1),x(2),...,x(k)} satisfies the property that all differences |x(i)-x(j)| are distinct (or alternately, all the sums are distinct), then it is called a Sidon set. So this sequence is basically the number of Sidon subsets of {1,2,...,n}. - Sayan Dutta, Feb 15 2024
See A143824 for sizes of the largest subsets of {1,2,...,n} with the desired property.
Also the number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum. - Gus Wiseman, Jun 07 2019

			{1,2,4} is a subset of {1,2,3,4}, with distinct differences 2-1=1, 4-1=3, 4-2=2 between pairs of elements, so {1,2,4} is counted as one of the 13 subsets of {1,2,3,4} with the desired property.  Only 2^4-13=3 subsets of {1,2,3,4} do not have this property: {1,2,3}, {2,3,4}, {1,2,3,4}.
From _Gus Wiseman_, May 17 2019: (Start)
The a(0) = 1 through a(5) = 22 subsets:
  {}  {}   {}     {}     {}       {}
      {1}  {1}    {1}    {1}      {1}
           {2}    {2}    {2}      {2}
           {1,2}  {3}    {3}      {3}
                  {1,2}  {4}      {4}
                  {1,3}  {1,2}    {5}
                  {2,3}  {1,3}    {1,2}
                         {1,4}    {1,3}
                         {2,3}    {1,4}
                         {2,4}    {1,5}
                         {3,4}    {2,3}
                         {1,2,4}  {2,4}
                         {1,3,4}  {2,5}
                                  {3,4}
                                  {3,5}
                                  {4,5}
                                  {1,2,4}
                                  {1,2,5}
                                  {1,3,4}
                                  {1,4,5}
                                  {2,3,5}
                                  {2,4,5}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100 (terms 0..60 from Alois P. Heinz, 61..81 from Vaclav Kotesovec)
Thomas Bloom, Problem 861, Erdős Problems.
Terence Tao, Erdős problem database, see no. 861.
Wikipedia, Sidon sequence.

First differences are A308251.
Second differences are A169942.
Row sums of A381476.
The maximal case is A325879.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A143824, A054578, A169947.
Cf. A000079, A108917, A143824, A169942, A308251, A325676, A325677, A325679, A325683, A325860, A325864, A241688, A241689, A241690.

b:= proc(n, s) local sn, m;
      if n<1 then 1
    else sn:= [s[], n];
         m:= nops(sn);
         `if`(m*(m-1)/2 = nops(({seq(seq(sn[i]-sn[j],
           j=i+1..m), i=1..m-1)})), b(n-1, sn), 0) +b(n-1, s)
      fi
    end:
a:= proc(n) option remember;
       b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 14 2011

b[n_, s_] := Module[{ sn, m}, If[n<1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m-1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m-1}, {j, i+1, m}] // Flatten // Union], b[n-1, sn], 0] + b[n-1, s]]]; a[n_] := a[n] = b[n - 1, {n}] + If[n == 0, 0, a[n-1]]; Table [a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Abs[Subtract@@@Subsets[#,{2}]]&]],{n,0,15}] (* Gus Wiseman, May 17 2019 *)

from itertools import combinations
def is_sidon_set(s):
    allsums = []
    for i in range(len(s)):
        for j in range(i, len(s)):
            allsums.append(s[i] + s[j])
    if len(allsums)==len(set(allsums)):
        return True
    return False
def a(n):
    sidon_count = 0
    for r in range(n + 1):
        subsets = combinations(range(1, n + 1), r)
        for subset in subsets:
            if is_sidon_set(subset):
                sidon_count += 1
    return sidon_count
print([a(n) for n in range(20)]) # Sayan Dutta, Feb 15 2024

from functools import cache
def b(n, s):
    if n < 1: return 1
    sn = s + [n]
    m = len(sn)
    return (b(n-1, sn) if m*(m-1)//2 == len(set(sn[i]-sn[j] for i in range(m-1) for j in range(i+1, m))) else 0) + b(n-1, s)
@cache
def a(n): return b(n-1, [n]) + (0 if n==0 else a(n-1))
print([a(n) for n in range(31)]) # Michael S. Branicky, Feb 15 2024 after Alois P. Heinz

a(n) = A169947(n-1) + n + 1 for n>=2. - Nathaniel Johnston, Nov 12 2010

a(n) = A054578(n) + 1 for n>0. - Alois P. Heinz, Jan 17 2013

a(21)-a(29) from Nathaniel Johnston, Nov 12 2010

Corrected a(21)-a(29) and more terms from Alois P. Heinz, Sep 14 2011

A241688 Number of Sidon subsets of {1,...,n} of size 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 10, 26, 60, 110, 190, 304, 466, 676, 958, 1312, 1762, 2310, 2984, 3786, 4750, 5874, 7196, 8720, 10484, 12488, 14780, 17360, 20276, 23530, 27174, 31210, 35696, 40630, 46074, 52032, 58566, 65676, 73434, 81840, 90966, 100814, 111460, 122906

Offset: 1

Carl Najafi, Apr 27 2014

nonn

A Sidon set is a set of natural numbers A={a_1,a_2,...}, finite or infinite, such that all pairwise sums a_i+a_j (i <= j) are different.

			a(7)=2 since the only subsets of {1,...,7} satisfying the required conditions are {1,2,5,7} and {1,3,6,7}.

Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1).

Column k=4 of A381476.

Cf. A054578.

SidonQ[l__] := If[Length[Join[Plus @@@ Subsets[l, {2}], 2 l]] == Length[Union[Join[Plus @@@ Subsets[l, {2}], 2 l]]], True, False]
Table[Length@Select[Subsets[Range[n], {4}], SidonQ[#] &], {n, 1, 30}]

It appears to be the case that G.f.: 2*x^7*(1+3*x+3*x^2+5*x^3)/((1-x)^5*(1+x)^2*(1+x^2)*(1+x+x^2)), corrected by Vaclav Kotesovec, May 03 2014
a(n) ~ 1/24*n^4 (deduced from g.f.). - Ralf Stephan, Apr 29 2014
a(n) = a(n-11)+a(n-8)-a(n-3)+2*(a(n-6)+a(n-1)-a(n-10)-a(n-5)). - Fung Lam, May 02 2014
Explicit formula (derived from g.f.): a(n) = n^4/24 - 7*n^3/12 + 29*n^2/12 - 15*n/8 - floor(n/4) - 4/3*floor(n/3) + (n/2-9/4)*floor(n/2) - floor((n+1)/4) - 2/3*floor((n+1)/3). - Vaclav Kotesovec, May 03 2014

A241689 Number of Sidon subsets of {1,...,n} of size 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 22, 68, 156, 320, 584, 1008, 1622, 2520, 3734, 5428, 7612, 10488, 14126, 18744, 24390, 31436, 39914, 50212, 62390, 76932, 93918, 113960, 137058, 163896, 194632, 229988, 270018, 315712, 367106, 425220, 490164, 563080, 644096

Offset: 1

Carl Najafi, Apr 27 2014

nonn

A Sidon set is a set of natural numbers A={a_1,a_2,...}, finite or infinite, such that all pairwise sums a_i+a_j (i <= j) are different.

			a(12)=4 since the only subsets of {1,...,12} satisfying the required conditions are {1,2,5,10,12}, {1,3,8,9,12}, {1,3,8,11,12}, and {1,4,5,10,12}.

Column k=5 of A381476.

Cf. A054578.

SidonQ[l__] := If[Length[Join[Plus @@@ Subsets[l, {2}], 2 l]] == Length[Union[Join[Plus @@@ Subsets[l, {2}], 2 l]]], True, False]
Table[Length@Select[Subsets[Range[n], {5}], SidonQ[#] &], {n, 1, 30}]

A241690 Number of Sidon subsets of {1,...,n} of size 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 24, 80, 206, 504, 1004, 1910, 3380, 5688, 9036, 14106, 21190, 31158, 44370, 62206, 85308, 115662, 153746, 202146, 262156, 336960, 427488, 538690, 671604, 831926, 1021238, 1246604, 1510056, 1820580, 2179480

Offset: 1

Carl Najafi, Apr 27 2014

nonn

A Sidon set is a set of natural numbers A={a_1,a_2,...}, finite or infinite, such that all pairwise sums a_i+a_j (i <= j) are different.

			a(18)=8 since there are 8 subsets of {1,...,18} satisfying the required conditions, for example {1,2,5,11,13,18}.

Column k=6 of A381476.

Cf. A054578.

SidonQ[l__] := If[Length[Join[Plus @@@ Subsets[l, {2}], 2 l]] == Length[Union[Join[Plus @@@ Subsets[l, {2}], 2 l]]], True, False]
Table[Length@Select[Subsets[Range[n], {6}], SidonQ[#] &], {n, 1, 30}]

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A143823 Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.

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A241688 Number of Sidon subsets of {1,...,n} of size 4.

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A241689 Number of Sidon subsets of {1,...,n} of size 5.

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A241690 Number of Sidon subsets of {1,...,n} of size 6.

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