OFFSET
1,1
COMMENTS
Column 3 of A207729.
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of formula
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-6,0,0,3,1,0,-1).
FORMULA
Empirical: a(n) = 3*a(n-1) - 2*a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-7) + a(n-8) - a(n-10).
Formula confirmed by Robert Israel, Mar 05 2018 (see link).
G.f.: x*(6 + 18*x - 18*x^2 + 9*x^3 + 7*x^4 + 3*x^5 - 9*x^6 - x^7 - x^8 + x^9) / ((1 - x)*(1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)). - Colin Barker, Mar 05 2018
EXAMPLE
Some solutions for n=4:
1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1
1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1
1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 1 0 0 1
1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 0 0 1
MAPLE
Configs:= select(A -> A[1..3] <> [0, 0, 0] and A[4..6] <> [0, 0, 0] and A[1..3] <> [0, 1, 0] and A[4..6] <> [0, 1, 0],
[seq(convert(x, base, 2)[1..6], x=2^6..2^7-1)]):
compat:= proc(i, j) local k, col;
if Configs[i][4..6] <> Configs[j][1..3] then return 0 fi;
for k from 1 to 3 do
col:= [Configs[i][k], Configs[i][k+3], Configs[j][k+3]];
if col = [0, 1, 1] or col = [1, 0, 1] then return 0 fi;
od;
1
end proc:
T:= Matrix(36, 36, compat):
u:= Vector[row](36, 1):
v:= Vector(36, 1):
6, seq(u . T^(n-2) . v, n=2..50); # Robert Israel, Mar 05 2018
MATHEMATICA
LinearRecurrence[{3, -2, 3, -6, 0, 0, 3, 1, 0, -1}, {6, 36, 78, 189, 490, 1113, 2449, 5474, 12036, 26100}, 30] (* Jean-François Alcover, May 15 2023, after Robert Israel's confirmed formula *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Feb 19 2012
STATUS
approved