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A325879 Number of maximal subsets of {1..n} such that every ordered pair of distinct elements has a different difference.

%I A325879 #26 Mar 27 2025 19:15:11
%S A325879 1,1,1,3,3,6,14,20,24,36,64,110,176,238,294,370,504,736,1086,1592,
%T A325879 2240,2982,3788,4700,5814,7322,9396,12336,16552,22192,29310,38046,
%U A325879 48368,60078,73722,89416,108208,131310,160624,198002,247408,310410,390924,490818,613344,758518
%N A325879 Number of maximal subsets of {1..n} such that every ordered pair of distinct elements has a different difference.
%C A325879 Also the number of maximal subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum.
%H A325879 Fausto A. C. Cariboni, <a href="/A325879/b325879.txt">Table of n, a(n) for n = 0..100</a>
%H A325879 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sidon_sequence">Sidon sequence</a>.
%H A325879 <a href="/index/Go#Golomb">Index entries for sequences related to Golomb rulers</a>.
%e A325879 The a(0) = 1 through a(7) = 20 subsets:
%e A325879   {}  {1}  {1,2}  {1,2}  {2,3}    {1,2,4}  {1,2,4}  {1,2,4}
%e A325879                   {1,3}  {1,2,4}  {1,2,5}  {1,2,5}  {1,2,6}
%e A325879                   {2,3}  {1,3,4}  {1,3,4}  {1,2,6}  {1,3,4}
%e A325879                                   {1,4,5}  {1,3,4}  {1,4,5}
%e A325879                                   {2,3,5}  {1,3,6}  {1,4,6}
%e A325879                                   {2,4,5}  {1,4,5}  {1,5,6}
%e A325879                                            {1,4,6}  {2,3,5}
%e A325879                                            {1,5,6}  {2,3,6}
%e A325879                                            {2,3,5}  {2,3,7}
%e A325879                                            {2,3,6}  {2,4,5}
%e A325879                                            {2,4,5}  {2,4,7}
%e A325879                                            {2,5,6}  {2,5,6}
%e A325879                                            {3,4,6}  {2,6,7}
%e A325879                                            {3,5,6}  {3,4,6}
%e A325879                                                     {3,4,7}
%e A325879                                                     {3,5,6}
%e A325879                                                     {4,5,7}
%e A325879                                                     {4,6,7}
%e A325879                                                     {1,2,5,7}
%e A325879                                                     {1,3,6,7}
%t A325879 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t A325879 Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]
%o A325879 (PARI)
%o A325879 a(n)={
%o A325879   my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1<<k)<<k), return(0))); 1);
%o A325879   my(recurse(k,b,w)=
%o A325879       if(k > n, ismaxl(b,w),
%o A325879          my(s=self()(k+1, b,w));
%o A325879          b+=1<<k; if(!bitand(w,b<<k), s+=self()(k+1, b, w + (b<<k)));
%o A325879          s);
%o A325879   );
%o A325879   recurse(1,0,0);
%o A325879 } \\ _Andrew Howroyd_, Mar 27 2025
%Y A325879 The subset case is A143823.
%Y A325879 The integer partition case is A325858.
%Y A325879 The strict integer partition case is A325876.
%Y A325879 Heinz numbers of the counterexamples are given by A325992.
%Y A325879 Cf. A002033, A108917, A143824, A196723, A382397.
%Y A325879 Cf. A325859, A325861, A325865, A325867, A325869, A325878, A325992.
%K A325879 nonn
%O A325879 0,4
%A A325879 _Gus Wiseman_, Jun 02 2019
%E A325879 a(21)-a(45) from _Fausto A. C. Cariboni_, Feb 08 2022