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A245094 Total squares count in n-th generation of Pythagoras tree variation which is rhombitrihexagonal tiling.

%I A245094 #69 Feb 16 2025 08:33:23
%S A245094 1,2,4,8,13,20,24,27,33,36,42,45,51,54,60,63,69,72,78,81,87,90,96,99,
%T A245094 105,108,114,117,123,126,132,135,141,144,150,153,159,162,168,171,177,
%U A245094 180,186,189,195,198,204,207,213,216,222,225,231,234,240,243,249,252,258,261,267
%N A245094 Total squares count in n-th generation of Pythagoras tree variation which is rhombitrihexagonal tiling.
%C A245094 Refer to Pythagoras tree (fractal) in the link. In the article "Varying the angle", the construction rule is changed from the standard Pythagoras tree by changing the base angle from 90 degrees to 60 degrees. It is easily seen that the size of the unit squares remains constant and equal to sin(30 degrees)/(1/2) = 1. The first overlap occurs at the fifth generation (n=4). The general pattern produced is the rhombitrihexagonal tiling, an array of hexagons bordered by the constructing squares. a(n) gives total count of squares in n-th generation which excluding the overlap into (n-1)-th generation and count only 1 for the overlap among current one. See illustration.
%C A245094 Conjecture: In the limit n -> infinity this construction produces one of the eight planar semiregular tessellations (one of the 11 Archimedean tessellations, the other three being regular). This is the tessellation (3,4,6,4) because of the sequence of regular 3-, 4- and 6-gons around each vertex. See the Eric Weisstein link. - _Wolfdieter Lang_, Nov 23 2014
%H A245094 Kival Ngaokrajang, <a href="/A245094/a245094.pdf">Illustration of initial terms</a>
%H A245094 John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>
%H A245094 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SemiregularTessellation.html">Semiregular Tessellation </a>
%H A245094 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagoras_tree_(fractal)">Pythagoras tree</a>
%H A245094 Wikipedia, <a href="http://en.wikipedia.org/wiki/Rhombitrihexagonal_tiling">Rhombitrihexagonal tiling</a>
%H A245094 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1).
%F A245094 Conjectures from _Colin Barker_, Nov 12 2014: (Start)
%F A245094 a(n) = 3*((-1)^n + 6*n-5)/4 for n > 5.
%F A245094 a(n) = a(n-1) + a(n-2) - a(n-3) for n > 8.
%F A245094 G.f.: (2*x^8 - 4*x^7 - x^6 + 3*x^5 + 3*x^4 + 3*x^3 + x^2 + x + 1) / ((x-1)^2*(x+1)).
%F A245094 (End)
%F A245094 It follows from the above conjecture that this sequence consists of interlaced polynomials for n > 5: a(2n) = 3*(3n-1) and a(2n+1) = 9*n. - _Avi Friedlich_, May 09 2015
%t A245094 a[n_] := a[n] = If[n <= 6, {1, 2, 4, 8, 13, 20, 24}[[n+1]], a[n-1] + 6 - 3 Mod[n, 2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Nov 24 2016, adapted from PARI *)
%o A245094 (PARI)
%o A245094 {a=24;print1("1, 2, 4, 8, 13, 20, ",a,", ");
%o A245094 for (n=7,100,if (Mod(n,2)==1,d1=3,d1=6);a=a+d1;print1(a,", "))}
%Y A245094 Cf. A226454, A227298.
%K A245094 nonn
%O A245094 0,2
%A A245094 _Kival Ngaokrajang_, Nov 12 2014