A188516 Number of nX2 binary arrays without the pattern 1 1 0 diagonally, vertically or horizontally.
4, 16, 49, 144, 400, 1089, 2916, 7744, 20449, 53824, 141376, 370881, 972196, 2547216, 6671889, 17472400, 45751696, 119793025, 313644100, 821166336, 2149898689, 5628600576, 14736017664, 38579637889, 101003196100, 264430435984
Offset: 1
Keywords
Examples
Some solutions for 3X2 ..0..1....0..1....0..0....0..0....1..0....0..1....1..0....0..1....0..0....0..1 ..0..0....0..0....0..0....0..1....1..1....1..0....0..1....0..1....1..0....1..0 ..1..1....0..0....0..1....1..0....1..1....0..0....1..0....1..1....0..0....0..1
Links
- R. H. Hardin, Table of n, a(n) for n = 1..200
Formula
Empirical: a(n)=4*a(n-1)-2*a(n-2)-6*a(n-3)+4*a(n-4)+2*a(n-5)-a(n-6).
Conjecture: a(n) = (F(n+3) - 1)^2, where F = A000045 (Fibonacci numbers). - Clark Kimberling, Jun 21 2016
Assuming the conjecture, define b(1) = 1 and b(n) = a(n-1) for n > 1. Then b(n) = Sum{F(i,j): (i=n and 1<=j<=n) or (j=n and 1<=i<=n)}, where F is the Fibonacci fusion array, A202453. - Clark Kimberling, Jun 21 2016
G.f. for (b(n)): -x*(-1+x^3-2*x^2) / ( (x-1)*(1+x)*(x^2-3*x+1)*(x^2+x-1) ). - R. J. Mathar, Dec 20 2011
b(n) = -2*(-1)^n/5 - 2*Fibonacci(n+2) + Lucas(2*n+4)/5 + 1. - Ehren Metcalfe, Mar 26 2016
Comments