OFFSET
0,11
COMMENTS
(Start) See A187502 for supporting theory. Define the matrix
U_2=
(0 0 1 0)
(0 1 0 1)
(1 0 1 1)
(0 1 1 1).
Let r>=0, and let A_r be the r-th "block" defined by A_r={a(3*r-2),a(3*r),a(3*r+1),a(3*r+2)} with a(-2)=1. Note that A_r-3*A_(r-1)+3*A_(r-3)={0,0,0,0}, for r>=4, with initial conditions {A_k}={{1,0,0,0},{0,0,1,0},{1,0,1,1},{1,1,3,2}}, k=0,1,2,3. Let p={p_1,p_2,p_3,p_4}={-2,0,1,2}, n=3*r+p_i and M=(U_2)^r. Then A_r corresponds component-wise to the first column of M, and a(n)=a(3*r+p_i)=m_(i,1) gives the quantity of H_(9,1,0) tiles that should appear in a subdivided H_(9,i,r) tile. (End)
Since a(3*r+1)=a(3*(r+1)-2) for all r, this sequence arises by concatenation of first-column entries m_(2,1), m_(3,1) and m_(4,1) from successive matrices M=(U_2)^r.
This sequence is a nontrivial extension of A187501.
REFERENCES
L. E. Jeffery, Unit-primitive matrices and rhombus substitution tilings, (in preparation).
LINKS
Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, 0, 0, 0, -3).
FORMULA
Recurrence: a(n)=3*a(n-3)-3*a(n-9), for n>=12, with initial conditions {a(m)}={0,0,0,0,1,0,0,1,1,1,3,2}, m=0,1,...,11.
G.f.: x^4(1-2*x^3+x^4+x^5-x^7)/(1-3*x^3+3*x^9).
MATHEMATICA
Join[{0, 0, 0}, LinearRecurrence[{0, 0, 3, 0, 0, 0, 0, 0, -3}, {0, 1, 0, 0, 1, 1, 1, 3, 2}, 50]] (* Harvey P. Dale, Jun 22 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
L. Edson Jeffery, Mar 16 2011
STATUS
approved