OFFSET
0,1
COMMENTS
a(n) appears to be sum of the elements of the n-th power of the matrix {{1, 2}, {2, 0}}. - Griffin N. Macris, Mar 25 2016
Let b(n) and c(n) be the number of such arrays where the two values in the bottom row are equal and not equal respectively so that a(n) = (1/2)*(b(n) + c(n)). Then b(n) and c(n) satisfy the recursive equations b(n+1) = b(n) + 2*c(n) and c(n+1) = 2*b(n). These equations correspond to the observation above and lead to an order 2 linear recurrence for a(n). - Andrew Howroyd, Jan 09 2025
LINKS
R. H. Hardin, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,4).
FORMULA
a(n) = a(n-1) + 4*a(n-2).
a(n) = (sqrt(17)/34)*((21 + 5*sqrt(17))*(1/2 + sqrt(17)/2)^(n-1) - (21 - 5*sqrt(17))*(1/2 - sqrt(17)/2)^(n-1)). - Taras Goy, Jan 04 2025
G.f.: (2 + 3*x)*(1 - x - 4*x^2). - Andrew Howroyd, Jan 09 2025
EXAMPLE
Some solutions with a(1,1)=0 for 3 X 2:
..0..1....0..0....0..1....0..1....0..1....0..0....0..0....0..1....0..0....0..0
..1..1....0..1....1..1....0..0....0..0....0..1....1..0....1..1....0..0....1..0
..1..1....0..0....1..0....1..0....0..0....1..1....1..1....0..1....0..0....0..0
PROG
(PARI) a(n)=vecsum([1, 1]*[1, 2; 2, 0]^n) \\ Andrew Howroyd, Jan 09 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Jan 07 2011
EXTENSIONS
a(0) = 2 prepended by Andrew Howroyd, Jan 09 2025
STATUS
approved