Edward Moody - cp's OEIS frontend

User: Edward Moody

Edward Moody has authored 2 sequences.

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

Edward Moody, Aug 02 2020

nonn

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.

			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.

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

Cf. A014552, A336861.

A336861 a(n) = ceiling((n-1-sqrt(n+1))/2).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32

Offset: 1

Edward Moody, Aug 16 2020

a(n) is a lower bound for the number of items outside the instance of n-1 at one end of a Colombian variant Langford pairing (A336747). For example, one of the most lop-sided pairings for n=7 is 4 1 6 1 7 4 3 5 2 6 3 2 7 5, and there are a(n)=2 items to the left of the first '6'. This bound is tight until at least n=184.

Edward Moody, Java program for enumerating Colombian Langford pairings

Cf. A336747.

[Ceiling((n-1-Sqrt(n+1))/2) : n in [1..100]]; // Wesley Ivan Hurt, Aug 21 2020

Table[Ceiling[(n - 1 - Sqrt[n + 1])/2], {n, 1, 100}] (* Amiram Eldar, Aug 21 2020 *)

a(n) = ceil((n-1-sqrt(n+1))/2); \\ Michel Marcus, Aug 19 2020

cp's OEIS Frontend

User: Edward Moody

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A336861 a(n) = ceiling((n-1-sqrt(n+1))/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI