login
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.
7
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.
Sequence in context: A056225 A343390 A105437 * A168140 A333699 A088100
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 10 2019
STATUS
approved