A054578 -id:A054578 - 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

nonn

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)
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

A169942 Number of Golomb rulers of length n.

Original entry on oeis.org

1, 1, 3, 3, 5, 7, 13, 15, 27, 25, 45, 59, 89, 103, 163, 187, 281, 313, 469, 533, 835, 873, 1319, 1551, 2093, 2347, 3477, 3881, 5363, 5871, 8267, 9443, 12887, 14069, 19229, 22113, 29359, 32229, 44127, 48659, 64789, 71167, 94625, 105699, 139119, 151145, 199657

Offset: 1

N. J. A. Sloane, Aug 01 2010

nonn

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
Leading entry in row n of triangle in A169940. Also the number of Sidon sets A with min(A) = 0 and max(A) = n. Odd for all n since {0,n} is the only symmetric Golomb ruler, and reversal preserves the Golomb property. Bounded from above by A032020 since the ruler {0 < r_1 < ... < r_t < n} gives rise to a composition of n: (r_1 - 0, r_2 - r_1, ... , n - r_t) with distinct parts. - Tomas Boothby, May 15 2012
Also the number of compositions of n such that every restriction to a subinterval has a different sum. This is a stronger condition than all distinct consecutive subsequences having a different sum (cf. A325676). - Gus Wiseman, May 16 2019

			For n=2, there is one Golomb Ruler: {0,2}.  For n=3, there are three: {0,3}, {0,1,3}, {0,2,3}. - _Tomas Boothby_, May 15 2012
From _Gus Wiseman_, May 16 2019: (Start)
The a(1) = 1 through a(8) = 15 compositions such that every restriction to a subinterval has a different sum:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)
            (12)  (13)  (14)  (15)   (16)   (17)
            (21)  (31)  (23)  (24)   (25)   (26)
                        (32)  (42)   (34)   (35)
                        (41)  (51)   (43)   (53)
                              (132)  (52)   (62)
                              (231)  (61)   (71)
                                     (124)  (125)
                                     (142)  (143)
                                     (214)  (152)
                                     (241)  (215)
                                     (412)  (251)
                                     (421)  (341)
                                            (512)
                                            (521)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..99
T. Pham, Enumeration of Golomb Rulers (Master's thesis), San Francisco State U., 2011.
Eric Weisstein's World of Mathematics, Golomb Ruler.
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to Golomb rulers

Related to thickness: A169940-A169954, A061909.
Related to Golomb rulers: A036501, A054578, A143823.
Row sums of A325677.
Cf. A000079, A103295, A103300, A108917, A143824, A325466, A325545, A325676, A325678, A325679, A325683, A325686.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,15}] (* Gus Wiseman, May 16 2019 *)

def A169942(n):
    R = QQ['x']
    return sum(1 for c in cartesian_product([[0, 1]]*n) if max(R([1] + list(c) + [1])^2) == 2)
[A169942(n) for n in range(1,8)]
# Tomas Boothby, May 15 2012

a(n) = A169952(n) - A169952(n-1) for n>1. - Andrew Howroyd, Jul 09 2017

a(15)-a(30) from Nathaniel Johnston, Nov 12 2011

a(31)-a(50) from Tomas Boothby, May 15 2012

A036501 Number of inequivalent Golomb rulers with n marks and shortest length.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

N. J. A. Sloane

From Gus Wiseman, May 31 2019: (Start)
A Golomb ruler of length n is a subset of {0..n} containing 0 and n and such that every pair of distinct terms has a different difference. For example, the a(2) = 1 through a(8) = 1 Golomb rulers are:
{0,1}
{0,1,3}
{0,1,4,6}
{0,1,4,9,11}
{0,2,7,8,11}
{0,1,4,10,12,17}
{0,1,4,10,15,17}
{0,1,8,11,13,17}
{0,1,8,12,14,17}
{0,1,4,10,18,23,25}
{0,1,7,11,20,23,25}
{0,2,3,10,16,21,25}
{0,2,7,13,21,22,25}
{0,1,11,16,19,23,25}
{0,1,4,9,15,22,32,34}
Also half the number of length-(n - 1) compositions of A003022(n) such that every consecutive subsequence has a different sum. For example, the a(2) = 1 through a(8) = 1 compositions are (A = 10):
(1)
(1,2)
(1,3,2)
(1,3,5,2)
(2,5,1,3)
(1,3,6,2,5)
(1,3,6,5,2)
(1,7,3,2,4)
(1,7,4,2,3)
(1,3,6,8,5,2)
(1,6,4,9,3,2)
(2,1,7,6,5,4)
(2,5,6,8,1,3)
(1,A,5,3,4,2)
(1,3,5,6,7,A,2)
(End)

Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
Distributed.Net, Project OGR
J. B. Shearer, Table of known optimal Golomb rulers
Eric Weisstein's World of Mathematics, Golomb Ruler
Index entries for sequences related to Golomb rulers

Cf. A003022, A039953, A054578, A108917, A143823, A169942, A270813, A325677, A325683.

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}]

A275672 Size of a largest subset of a regular cubic lattice of nnn points without repeated distances.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9

Offset: 0

Vladimir Reshetnikov, Aug 04 2016

10 <= a(7) <= 12, 11 <= a(8) <= 13, 12 <= a(9) <= 15, 13 <= a(10) <= 17.

			For n = 5, a(5) >= 7 is witnessed by {(1,1,1), (1,1,2), (1,1,4), (1,2,5), (2,3,1), (4,4,5), (5,5,4)}. There are 4223 distinct (up to rotation and reflection) 7-point configurations without repeated distances, and none of them can be extended to 8 points, so a(5) = 7.

Math.StackExchange, A largest subset of a cubic lattice with unique distances between its points, Aug 03 2016.
Ed Pegg Jr, No Repeated Distances, Wolfram Demonstrations Project, May 03 2013.
A. Zimmermann. Al Zimmermann's Programming Contests: Point Packing, Oct 10 2009.

Cf. A003022, A054578, A036501, A169942.

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.

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A169942 Number of Golomb rulers of length n.

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A036501 Number of inequivalent Golomb rulers with n marks and shortest length.

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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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A275672 Size of a largest subset of a regular cubic lattice of n*n*n points without repeated distances.

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A275672 Size of a largest subset of a regular cubic lattice of nnn points without repeated distances.