A301819 Number of nX4 0..1 arrays with every element equal to 1 or 2 horizontally or antidiagonally adjacent elements, with upper left element zero.

2, 8, 25, 68, 177, 457, 1183, 3075, 8018, 20947, 54782, 143348, 375200, 982178, 2571247, 6731462, 17623023, 46137475, 120789253, 316230117, 827900912, 2167472413, 5674516100, 14856075638, 38893710542, 101825055692, 266581456213

Offset

1,1

Comments

Column 4 of A301823.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Formula

Empirical: a(n) = 6*a(n-1) - 13*a(n-2) + 13*a(n-3) - 6*a(n-4) + a(n-5).

From Reda Belaiche, May 08 2026: (Start)

a(n) = 5*Fibonacci(2*n-1) + n*(n-1)/2 - 3.

The above recurrence follows from the closed form: Fibonacci(2*n-1) satisfies x^2 - 3*x + 1 = 0 and n*(n-1)/2 satisfies (x-1)^3 = 0, giving minimal polynomial x^5 - 6*x^4 + 13*x^3 - 13*x^2 + 6*x - 1. (End)

Example

Some solutions for n=5
..0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..1..1
..1..1..0..0. .0..1..0..0. .0..1..0..1. .0..0..0..0. .0..0..1..1
..0..0..1..1. .1..0..1..1. .1..0..1..0. .1..1..0..1. .0..0..1..0
..1..1..0..1. .0..1..0..0. .0..1..0..1. .1..0..1..1. .1..1..0..1
..0..0..1..1. .1..0..1..1. .1..0..1..1. .0..0..0..0. .0..0..1..1

Crossrefs

Cf. A301823.

Sequence in context: A370637 A066374 A193048 * A119854 A176855 A309237

Adjacent sequences: A301816 A301817 A301818 * A301820 A301821 A301822

Keyword

nonn

Author

R. H. Hardin, Mar 27 2018

Status

approved