This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A187069 #70 Aug 16 2025 17:29:48
%S A187069 0,1,0,1,1,2,2,4,5,9,11,20,25,45,56,101,126,227,283,510,636,1146,1429,
%T A187069 2575,3211,5786,7215,13001,16212,29213,36428,65641,81853,147494,
%U A187069 183922,331416,413269,744685,928607,1673292,2086561,3759853,4688460,8448313,10534874
%N A187069 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).
%C A187069 See A187070 for supporting theory. Define the matrix
%C A187069 U_2 = (0 0 1)
%C A187069 (0 1 1)
%C A187069 (1 1 1).
%C A187069 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.
%C A187069 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.
%H A187069 G. C. Greubel, <a href="/A187069/b187069.txt">Table of n, a(n) for n = 0..1000</a>
%H A187069 L. Edson Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A187069 Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H A187069 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1,0,-1).
%F A187069 Recurrence: a(n) = 2*a(n-2) + a(n-4) - a(n-6).
%F A187069 G.f.: x*(1-x^2+x^3-x^4)/(1-2*x^2-x^4+x^6).
%F A187069 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.
%F A187069 a(2*n) = A006054(n), a(2*n+3) = A052534(n).
%e A187069 Suppose r=3.
%e A187069 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
%e A187069 M = (U_2)^3 = (1 2 3)
%e A187069 (2 4 5)
%e A187069 (3 5 6).
%e A187069 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.
%t A187069 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 *)
%t A187069 LinearRecurrence[{0,2,0,1,0,-1},{0,1,0,1,1,2},50] (* _Harvey P. Dale_, Dec 16 2017 *)
%o A187069 (PARI) my(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
%Y A187069 Cf. A038196, A187068, A187070, A006054.
%K A187069 nonn,easy
%O A187069 0,6
%A A187069 _L. Edson Jeffery_, Mar 06 2011