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A187069
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Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i-1. Then a(n)=a(2*r+i-1) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).
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7
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0, 1, 0, 1, 1, 2, 2, 4, 5, 9, 11, 20, 25, 45, 56, 101, 126, 227, 283, 510, 636, 1146, 1429, 2575, 3211, 5786, 7215, 13001, 16212, 29213, 36428, 65641, 81853, 147494, 183922, 331416, 413269, 744685, 928607, 1673292, 2086561, 3759853, 4688460, 8448313, 10534874
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OFFSET
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0,6
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COMMENTS
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(Start) See A187070 for supporting theory. Define the matrix
U_2=
(0 0 1)
(0 1 1)
(1 1 1).
Let r>=0, and let B_r be the r-th "block" defined by B_r={a(2*r),a(2*r+1),a(2*r+2)}. Note that B_r-2*B_(r-1)-B_(r-2)+B_(r-3)={0,0,0}. Let n=2*r+i-1 and M=(m_(i,j))=(U_2)^r. Then B_r corresponds component-wise to the second column of M, and a(n)=a(2*r+i-1)=m_(i,2) gives the quantity of H_(7,2,0) tiles that should appear in a subdivided H_(7,i,r) tile. (End)
Since a(2*r+2)=a(2*(r+1)) for all r, this sequence arises by concatenation of second-column entries m_(1,2) and m_(2,2) from successive matrices M=(U_2)^r.
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LINKS
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FORMULA
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Recurrence: a(n) = 2*a(n-2) + a(n-4) - a(n-6).
G.f.: x*(1-x^2+x^3-x^4)/(1-2*x^2-x^4+x^6).
Closed-form: a(n) = (1/14)*[[X_1+Y_1*(-1)^(n-1)]*[(w_2)^2-(w_3)^2]*(w_1)^(n-1)+[X_2+Y_2*(-1)^(n-1)]*[(w_3)^2-(w_1)^2]*(w_2)^(n-1)+[X_3+Y_3*(-1)^(n-1)]*[(w_1)^2-(w_2)^2]*(w_3)^(n-1)], where w_k = sqrt[(2cos(k*Pi/7))^2-1], X_k = (w_k)^4-(w_k)^2+w_k-1 and Y_k = (w_k)^4+(w_k)^2-w_k-1, k=1,2,3.
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EXAMPLE
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(Start) Suppose r=3. Then
B_r = B_3 = {a(2*r,a(2*r+1),a(2*r+2)} = {a(6),a(7),a(8)} = {2,4,5},
corresponding to the entries in the second column of
M = (U_2)^3 =
(1 2 3)
(2 4 5)
(3 5 6).
Suppose i=2. Setting n=2*r+i-1, then a(n) = a(2*r+i-1) = a(6+2-1) = a(7) = m_(2,2) = 4. Hence a subdivided H_(7,2,3) tile should contain a(7) = m_(2,2) = 4 H_(7,2,0) tiles. (End)
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MATHEMATICA
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CoefficientList[Series[x*(1 - x^2 + x^3 - x^4)/(1 - 2*x^2 - x^4 + x^6), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
LinearRecurrence[{0, 2, 0, 1, 0, -1}, {0, 1, 0, 1, 1, 2}, 50] (* Harvey P. Dale, Dec 16 2017 *)
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PROG
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(PARI) x='x+O('x^50); concat([0], Vec(x*(1-x^2+x^3-x^4)/(1-2*x^2-x^4+x^6))) \\ G. C. Greubel, Oct 20 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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