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 A245094 Total squares count in n-th generation of Pythagoras tree variation which is rhombitrihexagonal tiling. 5
 1, 2, 4, 8, 13, 20, 24, 27, 33, 36, 42, 45, 51, 54, 60, 63, 69, 72, 78, 81, 87, 90, 96, 99, 105, 108, 114, 117, 123, 126, 132, 135, 141, 144, 150, 153, 159, 162, 168, 171, 177, 180, 186, 189, 195, 198, 204, 207, 213, 216, 222, 225, 231, 234, 240, 243, 249, 252, 258, 261, 267 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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 LINKS Kival Ngaokrajang, Illustration of initial terms John Riordan and N. J. A. Sloane, Correspondence, 1974 Eric Weisstein's World of Mathematics, Semiregular Tessellation Wikipedia, Pythagoras tree Wikipedia, Rhombitrihexagonal tiling FORMULA Conjectures from Colin Barker, Nov 12 2014: (Start) a(n) = 3*((-1)^n + 6*n-5)/4 for n > 5. a(n) = a(n-1) + a(n-2) - a(n-3) for n > 8. 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)). (End) 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 MATHEMATICA 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 *) PROG (PARI) {a=24; print1("1, 2, 4, 8, 13, 20, ", a, ", "); for (n=7, 100, if (Mod(n, 2)==1, d1=3, d1=6); a=a+d1; print1(a, ", "))} CROSSREFS Cf. A226454, A227298. Sequence in context: A328005 A186752 A030503 * A164486 A084684 A011907 Adjacent sequences:  A245091 A245092 A245093 * A245095 A245096 A245097 KEYWORD nonn AUTHOR Kival Ngaokrajang, Nov 12 2014 STATUS approved

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Last modified November 30 05:00 EST 2021. Contains 349418 sequences. (Running on oeis4.)