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

1,2

Links

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

Formula

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

Example

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}

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}]

Prog

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

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.

Sequence in context: A056225 A343390 A105437 * A168140 A333699 A088100

Adjacent sequences: A323949 A323950 A323951 * A323953 A323954 A323955

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 10 2019

Status

approved