OFFSET
0,7
COMMENTS
(Start) See A187506 for supporting theory. Define the matrix
U_3=
(0 0 0 1)
(0 0 1 1)
(0 1 1 1)
(1 1 1 1).
2. Let r>=0 and M=(m_(i,j))=(U_3)^r, i,j=1,2,3,4. Let C_r be the r-th "block" defined by C_r={a(3*r-1),a(3*r),a(3*r+1),a(3*r+2)} with a(-1)=0. Note that C_r-2*C_(r-1)-3*C_(r-2)+C_(r-3)+C_(r-4)={0,0,0,0}, for r>=4. Let p={p_1,p_2,p_3,p_4}={-1,0,1,2} and n=3*r+p_i. Then a(n)=a(3*r+p_i)=m_(i,3), where M=(m_(i,j))=(U_3)^r was defined above. Hence the block C_r corresponds component-wise to the third column of M, and a(3*r+p_i)=m_(i,3) gives the quantity of H_(9,3,0) tiles that should appear in a subdivided H_(9,i,r) tile. (End)
Since a(3*r+2)=a(3*(r+1)-1) for all r, this sequence arises by concatenation of third-column entries m_(2,3), m_(3,3) and m_(4,3) of M=(U_3)^r.
FORMULA
Recurrence: a(n)=2*a(n-3)+3*a(n-6)-a(n-9)-a(n-12), for n>=12, with initial conditions {a(k)}={0,1,0,1,1,1,2,3,3,6,8,9}, k=0,1,...,11.
G.f.: x*(1+x^2-x^3+x^4-2*x^6+x^7-x^8)/(1-2*x^3-3*x^6+x^9+x^12).
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
L. Edson Jeffery, Mar 14 2011
STATUS
approved