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%I #21 Jun 04 2021 14:44:37
%S 1,3,6,6,8,13,12,14,15,17,22,21,23,25,28,29,29,32,36,37,36,42,42,40,
%T 45,47,51,51,51,55,57,57,58,58,69,66,63,73,73,72,72,72,84,75,84,88,85,
%U 87,86,91,98,94,96,100,107,103,100,105,113,110,110,115,115,114
%N a(n) is the number of (connected) tiles at distance n from the leftmost tile in the Hofstetter-4fold tiling.
%C We build the Hofstetter-4fold tiling as follows:
%C - H_0 corresponds to a 2 X 2 square:
%C +---+---+
%C | |
%C + +
%C | |
%C +---+---+
%C O
%C - for any k >= 0, H_{k+1} is obtained by arranging 4 copies of H_k, rotated by 0, 90, 180, 270 degrees clockwise respectively, as follows:
%C +.......+
%C . 90.
%C +.......+ +.......+ ....+
%C . . .0 . . .
%C . . --> . ..... .
%C . . . . 180.
%C +.......+ +.......+.......+
%C O O .270 .
%C +.......+
%C - note that:
%C - the copy rotated by 0 degrees hides some squares on the copies rotated by 90 and 270 degrees,
%C - the copy rotated by 90 degrees hides some squares on the copy rotated by 180 degrees,
%C - the copy rotated by 180 degrees hides some squares on the copy rotated by 270 degrees,
%C - the Hofstetter-4fold tiling corresponds to the limit of H_k as k tends to infinity,
%C - in this sequence we consider only connected tiles (whose squares are vertically or horizontally adjacent); disconnected tiles made up of two diagonally adjacent squares are counted as two distinct connected tiles.
%H Rémy Sigrist, <a href="/A342425/b342425.txt">Table of n, a(n) for n = 0..5000</a>
%H Rémy Sigrist, <a href="/A342425/a342425.png">Illustration of initial terms</a>
%H Rémy Sigrist, <a href="/A342425/a342425_1.png">Colored representation of the tiles at distance <= 512</a> (where the color is function of the distance)
%H Rémy Sigrist, <a href="/A342425/a342425.txt">C# program for A342425</a>
%H Tilings Encyclopedia, <a href="https://tilings.math.uni-bielefeld.de/substitution/hofstetter-4fd-plain/">Hofstetter-4fold (plain)</a>
%e See illustration in Links section.
%o (C#) See Links section.
%Y Cf. A342577, A342597.
%K nonn
%O 0,2
%A _Rémy Sigrist_, Mar 11 2021