This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306357 #12 Jan 06 2020 16:52:38
%S A306357 0,1,3,6,10,20,38,70,130,240,442,814,1498,2756,5070,9326,17154,31552,
%T A306357 58034,106742,196330,361108,664182,1221622,2246914,4132720,7601258,
%U A306357 13980894,25714874,47297028,86992798,160004702,294294530,541292032,995591266,1831177830
%N A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.
%C A306357 Cyclically successive means 1 is a successor of n.
%C A306357 Set partitions using these subsets are counted by A323949.
%F A306357 For n >= 3 we have a(n) = A001644(n) - 1.
%F A306357 From _Chai Wah Wu_, Jan 06 2020: (Start)
%F A306357 a(n) = 2*a(n-1) - a(n-4) for n > 6.
%F A306357 G.f.: x*(x^5 + x^4 - 2*x^3 + x + 1)/(x^4 - 2*x + 1). (End)
%e A306357 The a(1) = 1 through a(5) = 20 stable subsets:
%e A306357 {1} {1} {1} {1} {1}
%e A306357 {2} {2} {2} {2}
%e A306357 {1,2} {3} {3} {3}
%e A306357 {1,2} {4} {4}
%e A306357 {1,3} {1,2} {5}
%e A306357 {2,3} {1,3} {1,2}
%e A306357 {1,4} {1,3}
%e A306357 {2,3} {1,4}
%e A306357 {2,4} {1,5}
%e A306357 {3,4} {2,3}
%e A306357 {2,4}
%e A306357 {2,5}
%e A306357 {3,4}
%e A306357 {3,5}
%e A306357 {4,5}
%e A306357 {1,2,4}
%e A306357 {1,3,4}
%e A306357 {1,3,5}
%e A306357 {2,3,5}
%e A306357 {2,4,5}
%t A306357 stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
%t A306357 Table[Length[stabsubs[Partition[Range[n],3,1,1]]],{n,15}]
%Y A306357 Cf. A000126, A000296, A001610, A001644, A169985, A306357, A323949, A323952, A323955.
%K A306357 nonn
%O A306357 0,3
%A A306357 _Gus Wiseman_, Feb 10 2019