A336747 Number of Colombian variant Langford pairings (solutions to Langford problem such that exactly one instance of {1, 2, 3, ..., n-2, n} occurs between the two instances of n-1), up to reversal of the order.

0, 0, 1, 1, 0, 0, 3, 10, 0, 0, 76, 140, 0, 0, 2478, 5454, 0, 0, 105704, 267312, 0, 0, 7235244, 25244832, 0, 0, 709868768, 2310292004, 0, 0, 91242419796, 339602328050, 0, 0, 15469115987732, 54988746724416, 0, 0, 3075508960864496, 11965953308933012

Offset: 1

Author

Edward Moody, Aug 02 2020

Keywords

• nonn

Comments

The variant was devised by Bernardo Recamán Santos and Freddy Barrera in Bogotá, who calculated up to a(20).

Ceiling((n-1-sqrt(n+1))/2) is a lower bound for the number of items outside the instance of n-1 at one end, e.g. for n=7 there are at least two items before the first '6'. This bound is tight until at least n=184.

Examples

			The unique Langford pairings for n=3 and n=4 are also Colombian:
  3 1 2 1 3 2 and 4 1 3 1 2 4 3 2.
For n=7, the a(7)=3 solutions are:
  4 1 6 1 7 4 3 5 2 6 3 2 7 5,
  2 3 6 2 7 3 4 5 1 6 1 4 7 5,
  7 3 1 6 1 3 4 5 7 2 6 4 2 5.
		

Links

• Edward Moody, Table of n, a(n) for n = 1..66
• J. E. Miller, Colombian Variant of Langford's Problem
• Edward Moody, Java program for enumerating Colombian Langford pairings

Crossrefs

Cf. A014552, A336861.