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

1,7

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.

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

Example

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.

Crossrefs

Cf. A014552, A336861.

Sequence in context: A113116 A137044 A188545 * A212361 A011998 A278759

Adjacent sequences: A336744 A336745 A336746 * A336748 A336749 A336750

Keyword

nonn

Author

Edward Moody, Aug 02 2020

Status

approved