

A187070


Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i1. Then a(n)=a(2*r+i1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^21).


10



0, 0, 1, 1, 1, 2, 3, 5, 6, 11, 14, 25, 31, 56, 70, 126, 157, 283, 353, 636, 793, 1429, 1782, 3211, 4004, 7215, 8997, 16212, 20216, 36428, 45425, 81853, 102069, 183922, 229347, 413269, 515338, 928607, 1157954, 2086561, 2601899
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OFFSET

0,6


COMMENTS

Theory. (Start)
1. Definitions. Let T_(7,j,0) denote the rhombus with sides of unit length (=1), interior angles given by the pair (j*Pi/7,(7j)*Pi/7) and Area(T_(7,j,0))=sin(j*Pi/7), j in {1,2,3}. Associated with T_(7,j,0) are its angle coefficients (j, 7j) in which one coefficient is even while the other is odd. A halftile is created by cutting T_(7,j,0) along a line extending between its two corners with even angle coefficient; let H_(7,j,0) denote this halftile. Similarly, a T_(7,j,r) tile is a linearly scaled version of T_(7,j,0) with sides of length x^r and Area(T_(7,j,r))=x^(2*r)*sin(j*Pi/7), r>=0 an integer, where x is the positive, constant square root x=sqrt[(2*cos(j*Pi/7))^2  1]; likewise let H_(7,j,r) denote the corresponding halftile. Often H_(7,i,r) (i in {1,2,3}) can be subdivided into an integral number of each equivalence class H_(7,j,0). But regardless of whether or not H_(7,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, in which the entry m_(i,j) gives the quantity of H_(7,j,0) tiles that should be present in a subdivided H_(7,i,r) tile. The number x^(2*r) (the square of the scaling factor) is an eigenvalue of M=(U_2)^r, where
U_2=
(0 0 1)
(0 1 1)
(1 1 1).
2. The sequence. Let r>=0, and let C_r be the rth "block" defined by C_r={a(2*r),a(2*r+1),a(2*r+2)}. Note that C_r2*C_(r1)C_(r2)+C_(r3)={0,0,0}. Let n=2*r+i1. Then a(n)=a(2*r+i1)=m_(i,3), where M=(m_(i,j))=(U_2)^r was defined above. Hence the block C_r corresponds componentwise to the third column of M, and a(n)=m_(i,3) gives the quantity of H_(7,3,0) tiles that should appear in a subdivided H_(7,i,r) tile. (End)
Combining blocks A_r, B_r and C_r, from A187068, A187069 and this sequence, respectively, as matrix columns [A_r,B_r,C_r] generates the matrix (U_2)^r, and a negative index (1)*r yields the corresponding inverse [A_(r),B_(r),C_(r)]=(U_2)^(r) of (U_2)^r. Therefore, the three sequences need not be causal.
Since a(2*r+2)=a(2*(r+1)) for all r, this sequence arises by concatenation of thirdcolumn entries m_(1,3) and m_(2,3) from successive matrices M=(U_2)^r.
This sequence is a trivial extension of A038196.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unitprimitive matrices
R. Witula, D. Slota and A. Warzynski, QuasiFibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,1).


FORMULA

Recurrence: a(n) = 2*a(n2) + a(n4)  a(n6).
G.f.: x^2*(1+xx^2)/(12*x^2x^4+x^6).
a(2*n)=A106803(n); a(2*n+1)=A006054(n+1); a(2*n+2)=A077998(n).
Closedform: a(n) = (1/14)*[[X_1+Y_1*(1)^(n1)]*[(w_2)^2(w_3)^2]*(w_1)^(n1)+[X_2+Y_2*(1)^(n1)]*[(w_3)^2(w_1)^2]*(w_2)^(n1)+[X_3+Y_3*(1)^(n1)]*[(w_1)^2(w_2)^2]*(w_3)^(n1)], where w_k = sqrt[(2cos(k*Pi/7))^21], X_k = (w_k)^3+(w_k)^2w_k and Y_k = (w_k)^3+(w_k)^2+w_k, k=1,2,3.


EXAMPLE

Suppose r=3. Then
C_r = C_3 = {a(2*r,a(2*r+1),a(2*r+2)} = {a(6),a(7),a(8)} = {3,5,6},
corresponding to the entries in the third column of
M = (U_2)^3 =
(1 2 3)
(2 4 5)
(3 5 6).
Choose i=2 and set n=2*r+i1. Then a(n) = a(2*r+i1) = a(6+21) = a(7) = 5, which equals the entry in row 2 and column 3 of M. Hence a subdivided H_(7,2,3) tile should contain a(7) = m_(2,3) = 5 H_(7,3,0) tiles.


MATHEMATICA

a[0] = a[1] = 0; a[2] = a[3] = a[4] = 1; a[_?Negative] = 0; a[n_] := a[n] = 2*a[n2] + a[n4]  a[n6]; Table[a[n], {n, 0, 40}] (* JeanFrançois Alcover, Jan 02 2013 *)


PROG

(PARI) x='x+O('x^50); concat([0, 0], Vec(x^2*(1+xx^2)/(12*x^2x^4+x^6))) \\ G. C. Greubel, Jul 05 2017


CROSSREFS

Cf. A038196, A187068, A187069.
Sequence in context: A039844 A276107 A187068 * A038196 A039849 A039896
Adjacent sequences: A187067 A187068 A187069 * A187071 A187072 A187073


KEYWORD

nonn,easy


AUTHOR

L. Edson Jeffery, Mar 05 2011


STATUS

approved



