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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

Table of n, a(n) for n=0..60.

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.)