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A085489 a(n) is the number of subsets of {1,...,n} containing no solutions to x+y=z with x and y distinct (one version of "sum-free subsets").

%I A085489 #41 Feb 16 2025 08:32:50
%S A085489 1,2,4,7,13,22,37,61,102,162,261,410,646,1001,1553,2370,3645,5515,
%T A085489 8303,12470,18713,27811,41244,60962,89733,131870,192522,281125,408680,
%U A085489 593880,855661,1238592,1779614,2563476,3660084,5255913,7473380,10696444,15137517
%N A085489 a(n) is the number of subsets of {1,...,n} containing no solutions to x+y=z with x and y distinct (one version of "sum-free subsets").
%C A085489 First differs from A151897 at a(7) = 61, A151897(7) = 60. The one subset counted under a(7) but not under A151897(7) is {1,2,4,7}. - _Gus Wiseman_, Jun 07 2019
%H A085489 Fausto A. C. Cariboni, <a href="/A085489/b085489.txt">Table of n, a(n) for n = 0..75</a>, (terms up to a(57) from Ben Burns)
%H A085489 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Sum-FreeSet.html">Sum-Free Set</a> [Strictly speaking this link is not relevant, since it uses a different definition of "sum-free".]
%F A085489 a(n) = 2^n - A088809(n). - _Reinhard Zumkeller_, Oct 19 2003
%e A085489 From _Gus Wiseman_, Jun 07 2019: (Start)
%e A085489 The a(0) = 1 through a(4) = 13 subsets:
%e A085489   {}  {}   {}     {}     {}
%e A085489       {1}  {1}    {1}    {1}
%e A085489            {2}    {2}    {2}
%e A085489            {1,2}  {3}    {3}
%e A085489                   {1,2}  {4}
%e A085489                   {1,3}  {1,2}
%e A085489                   {2,3}  {1,3}
%e A085489                          {1,4}
%e A085489                          {2,3}
%e A085489                          {2,4}
%e A085489                          {3,4}
%e A085489                          {1,2,4}
%e A085489                          {2,3,4}
%e A085489 The a(5) = 22 subsets:
%e A085489   {}  {1}  {1,2}  {1,2,4}
%e A085489       {2}  {1,3}  {1,2,5}
%e A085489       {3}  {1,4}  {1,3,5}
%e A085489       {4}  {1,5}  {2,3,4}
%e A085489       {5}  {2,3}  {2,4,5}
%e A085489            {2,4}  {3,4,5}
%e A085489            {2,5}
%e A085489            {3,4}
%e A085489            {3,5}
%e A085489            {4,5}
%e A085489 (End)
%t A085489 Table[Length[Select[Subsets[Range[n]],Intersection[ #,Select[ Plus@@@ Subsets[ #,{2}],#<=n&]]=={}&]],{n,0,10}] (* _Gus Wiseman_, Jun 07 2019 *)
%Y A085489 See A007865 for another version.
%Y A085489 Cf. A050291, A051026, A103580, A151897, A308546, A326080, A326083, A326117.
%K A085489 nonn,nice
%O A085489 0,2
%A A085489 _Eric W. Weisstein_, Jul 02 2003
%E A085489 More terms from _Reinhard Zumkeller_, Jul 13 2003
%E A085489 Edited by _David Wasserman_, Apr 16 2008
%E A085489 a(0) = 1 prepended by _Gus Wiseman_, Jun 07 2019