%I
%S 0,0,1,1,1,1,2,3,4,7,9,10,19,26,30,56,75,85,160,216,246,462,622,707,
%T 1329,1791,2037,3828,5157,5864,11021,14849,16886,31735,42756,48620,
%U 91376,123111,139997,263108,354484,403104,757588,1020696,1160693,2181389
%N Let i be in {1,2,3,4} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3, p_4} = {1,0,1,2}, n=3*r+p_i, and define a(1)=0. Then a(n)=a(3*r+p_i) gives the quantity of H_(9,4,0) tiles in a subdivided H_(9,i,r) tile after linear scaling by the factor Q^r, where Q=sqrt(x^32*x) with x=2*cos(Pi/9).
%C Theory. (Start)
%C 1. Definitions. Let T_(9,j,0) denote the rhombus with sides of unit length (=1), interior angles given by the pair (j*Pi/9,(9j)*Pi/9) and Area(T_(9,j,0))=sin(j*Pi/9), j in {1,2,3,4}. Associated with T_(9,j,0) are its angle coefficients (j, 9j) in which one coefficient is even while the other is odd. A halftile is created by cutting T_(9,j,0) along a line extending between its two corners with even angle coefficient; let H_(9,j,0) denote this halftile. Similarly, a T_(9,j,r) tile is a linearly scaled version of T_(9,j,0) with sides of length Q^r and Area(T_(9,j,r))=Q^(2*r)*sin(j*Pi/9), r>=0 an integer, where Q is the positive, constant square root Q=sqrt(x^32*x) with x=2*cos(Pi/9); likewise let H_(9,j,r) denote the corresponding halftile. Often H_(9,i,r) (i in {1,2,3,4}) can be subdivided into an integral number of each equivalence class H_(9,j,0). But regardless of whether or not H_(9,j,r) subdivides, in theory such a proposed subdivision for each j can be represented by the matrix M=(m_(i,j)), i,j=1,2,3,4, in which the entry m_(i,j) gives the quantity of H_(9,j,0) tiles that should be present in a subdivided H_(9,i,r) tile. The number Q^(2*r) (the square of the scaling factor) is an eigenvalue of M=(U_3)^r, where
%C U_3=
%C (0 0 0 1)
%C (0 0 1 1)
%C (0 1 1 1)
%C (1 1 1 1).
%C 2. The sequence. Let r>=0, and let D_r be the rth "block" defined by D_r={a(3*r1),a(3*r),a(3*r+1),a(3*r+2)} with a(1)=0. Note that D_r2*D_(r1)3*D_(r2)+D_(r3)+D_(r4)={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,4), where M=(m_(i,j))=(U_3)^r was defined above. Hence the block D_r corresponds componentwise to the fourth column of M, and a(3*r+p_i)=m_(i,4) gives the quantity of H_(9,4,0) tiles that should appear in a subdivided H_(9,i,r) tile. (End)
%C Combining blocks A_r, B_r, C_r and D_r, from A187503, A187504, A187505 and this sequence, respectively, as matrix columns [A_r,B_r,C_r,D_r] generates the matrix (U_3)^r, and a negative index (1)*r yields the corresponding inverse [A_(r),B_(r),C_(r),D(r)]=(U_3)^(r) of (U_3)^r. Therefore, the four sequences need not be causal.
%C Since U_3 is symmetric, so is M=(U_3)^r, so the block D_r also corresponds to the fourth row of M. Therefore, alternatively, for j=1,2,3,4, a(3r+p_j)=m_(4,j) gives the quantity of H_(9,j,0) tiles that should be present in a H_(9,4,r) tile.
%C Since a(3*r+2)=a(3*(r+1)1) for all r>0, this sequence arises by concatenation of fourthcolumn entries m_(2,4), m_(3,4) and m_(4,4) (or fourthrow entries m_(4,2), m_(4,3) and m_(4,4)) from successive matrices M=(U_2)^r.
%C This sequence is a nontrivial extension of A038197.
%D L. E. Jeffery, Unitprimitive matrices and rhombus substitution tilings, (in preparation).
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,3,3,3,0,0,0,1,1,1).
%F Recurrence: a(n)=2*a(n3)+3*a(n6)a(n9)a(n12), with initial conditions {a(m)}={0,0,1,1,1,2,3,4,7,10,19,26}, m=0,1,...,11.
%F G.f.: x^2*(1+x+x^2x^3+x^5x^6)/(12*x^33*x^6+x^9+x^12).
%t LinearRecurrence[{1,1,3,3,3,0,0,0,1,1,1},{0,0,1,1,1,1,2,3,4,7,9},50] (* _Harvey P. Dale_, May 19 2015 *)
%Y Cf. A038197, A187503, A187504, A187505.
%K nonn,easy
%O 0,7
%A _L. Edson Jeffery_, Mar 10 2011
