A261478 Number of set partitions of [n] such that no part contains two elements with a circular distance less than three.

1, 1, 1, 1, 1, 1, 8, 29, 106, 491, 2449, 12860, 72488, 435241, 2763053, 18485280, 129916333, 956237591, 7351602714, 58897588844, 490680801682, 4242904633903, 38014082900386, 352341757997443, 3373662297796822, 33326335447469262, 339232538360853201

Offset

0,7

Comments

The circular distance of 1 and n is 1 (for n>1).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Example

a(0) = 1: {}.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 1: 1|2|3.
a(4) = 1: 1|2|3|4.
a(5) = 1: 1|2|3|4|5.
a(6) = 8: 14|25|36, 14|25|3|6, 14|2|36|5, 14|2|3|5|6, 1|25|36|4, 1|25|3|4|6, 1|2|36|4|5, 1|2|3|4|5|6.
a(7) = 29: 14|25|36|7, 14|25|37|6, 14|25|3|6|7, 14|26|37|5, 14|26|3|5|7, 14|2|36|5|7, 14|2|37|5|6, 14|2|3|5|6|7, 15|26|37|4, 15|26|3|47, 15|26|3|4|7, 15|2|36|47, 15|2|36|4|7, 15|2|37|4|6, 15|2|3|47|6, 15|2|3|4|6|7, 1|25|36|47, 1|25|36|4|7, 1|25|37|4|6, 1|25|3|47|6, 1|25|3|4|6|7, 1|26|37|4|5, 1|26|3|47|5, 1|26|3|4|5|7, 1|2|36|47|5, 1|2|36|4|5|7, 1|2|37|4|5|6, 1|2|3|47|5|6, 1|2|3|4|5|6|7.

Maple

b:= proc(n, l, m, h) option remember;
      `if`(n=0, `if`(1 in [l, m] or l=2, 0, 1), add(
      `if`(j in [l, m], 0, b(n-1, j, l, max(h, j))), j=1..h+1))
    end:
a:= n-> `if`(n<6, 1, b(n, 0$3)):
seq(a(n), n=0..30);

Mathematica

b[n_, l_, m_, h_] := b[n, l, m, h] = If[n==0, If[l==1 || m==1 || l==2, 0, 1], Sum[If[j==l || j==m, 0, b[n - 1, j, l, Max[h, j]]], {j, 1, h + 1}]];
a[n_] := If[n < 6, 1, b[n, 0, 0, 0]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Crossrefs

Row sums of A261477.

Sequence in context: A001360 A294838 A116952 * A199207 A088131 A072264

Adjacent sequences: A261475 A261476 A261477 * A261479 A261480 A261481

Keyword

nonn

Author

Alois P. Heinz, Aug 20 2015

Status

approved