A153367 Number of zig-zag paths from top to bottom of a rectangle of width 9 with 2n-1 rows whose color is not that of the top right corner.
4, 14, 50, 180, 650, 2350, 8500, 30750, 111250, 402500, 1456250, 5268750, 19062500, 68968750, 249531250, 902812500, 3266406250, 11817968750, 42757812500, 154699218750, 559707031250, 2025039062500, 7326660156250, 26508105468750
Offset: 1
Links
- Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Formula
Empirical g.f.: x*(4-6*x)/(1-5*x+5*x^2). - Colin Barker, Jan 04 2012
Conjectures from Colin Barker, Feb 11 2018: (Start)
a(n) = (2^(-n)*((5-sqrt(5))^n*(-5+3*sqrt(5)) + (5+sqrt(5))^n*(5+3*sqrt(5)))) / (5*sqrt(5)) for n>0.
a(n) = 5*a(n-1) - 5*a(n-2) for n>2.
(End)
Assuming Colin Barker's conjectures, a(2*n) = 2*5^(n-1)*Lucas(2*(n+1)), a(2*n+1) = 2*5^n*Fibonacci(2*n+3). - Ehren Metcalfe, Apr 21 2018