A323952 Regular triangle read by rows where if k > 1 then T(n, k) is the number of connected subgraphs of an n-cycle with any number of vertices other than 2 through k - 1, n >= 1, 1 <= k <= n - 1. Otherwise T(n, 1) = n.

1, 2, 3, 3, 7, 4, 4, 13, 9, 5, 5, 21, 16, 11, 6, 6, 31, 25, 19, 13, 7, 7, 43, 36, 29, 22, 15, 8, 8, 57, 49, 41, 33, 25, 17, 9, 9, 73, 64, 55, 46, 37, 28, 19, 10, 10, 91, 81, 71, 61, 51, 41, 31, 21, 11, 11, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 12, 133, 121

Offset: 1

Author

Gus Wiseman, Feb 10 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   1
   2   3
   3   7   4
   4  13   9   5
   5  21  16  11   6
   6  31  25  19  13   7
   7  43  36  29  22  15   8
   8  57  49  41  33  25  17   9
   9  73  64  55  46  37  28  19  10
  10  91  81  71  61  51  41  31  21  11
  11 111 100  89  78  67  56  45  34  23  12
  12 133 121 109  97  85  73  61  49  37  25  13
Row 4 counts the following connected sets:
  {1}  {1}     {1}     {1}
  {2}  {2}     {2}     {2}
  {3}  {3}     {3}     {3}
  {4}  {4}     {4}     {4}
       {12}    {123}   {1234}
       {14}    {124}
       {23}    {134}
       {34}    {234}
       {123}   {1234}
       {124}
       {134}
       {234}
       {1234}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

First column is A000027. Second column is A002061. Third column is A000290. Fourth column is A028387.

Cf. A000126, A000325, A306351, A323950, A323951, A323953, A323954, A323956.

Programs

Mathematica

anesw[n_,k_]:=Length[If[k==1,List/@Range[n],Union[Sort/@Select[Union[List/@Range[n],Join@@Table[Partition[Range[n],i,1,1],{i,k,n}]],UnsameQ@@#&&#!={}&]]]];
Table[anesw[n,k],{n,0,16},{k,n}]

PARI

T(n,k) = if(k==1, n, 1 + n * (n - k + 1)) \\ Andrew Howroyd, Jan 18 2023

Formula

T(n, 1) = n; T(n, k) = 1 + n * (n - k + 1).