A306351 Number of ways to split an n-cycle into connected subgraphs all having at least 4 vertices.

1, 0, 0, 0, 1, 1, 1, 1, 5, 10, 16, 23, 35, 53, 78, 111, 157, 222, 313, 438, 610, 848, 1178, 1634, 2263, 3131, 4330, 5986, 8272, 11427, 15782, 21794, 30093, 41548, 57359, 79183, 109307, 150887, 208279, 287496, 396838, 547761, 756077, 1043611, 1440488, 1988289

Offset: 0

Author

Gus Wiseman, Feb 10 2019

Keywords

• nonn
• easy

Examples

			The a(7) = 1 through a(9) = 10 partitions:
  {{1234567}}  {{12345678}}    {{123456789}}
               {{1234}{5678}}  {{1234}{56789}}
               {{1238}{4567}}  {{12345}{6789}}
               {{1278}{3456}}  {{12349}{5678}}
               {{1678}{2345}}  {{12389}{4567}}
                               {{1239}{45678}}
                               {{12789}{3456}}
                               {{1289}{34567}}
                               {{16789}{2345}}
                               {{1789}{23456}}
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-2,1).

Crossrefs

Column k = 3 of A323954.

Cf. A000325, A005251, A066982, A078012, A323950, A323951, A323953.

Programs

Mathematica

cycedsprop[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,i+k,n+i-1}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cycedsprop[n,3],Range[n]]],{n,15}]

Formula

G.f.: (2*x^9-3*x^8+x^3-3*x^2+3*x-1)/((x^4+x-1)*(x-1)^2). - Alois P. Heinz, Feb 10 2019

Extensions

More terms from Alois P. Heinz, Feb 10 2019