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%I #31 Jun 04 2021 14:44:42
%S 1,3,4,6,9,11,11,13,15,19,16,20,25,25,27,27,29,35,30,34,41,41,39,41,
%T 47,45,44,48,57,53,57,55,57,67,56,62,73,71,67,69,73,79,68,76,89,83,87,
%U 83,93,89,86,90,105,99,95,97,109,99,100,104,121,109,117,111
%N a(n) is the number of (not necessarily 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 connected tiles (whose squares are vertically or horizontally adjacent) as well as disconnected tiles (made up of two diagonally adjacent squares).
%H Rémy Sigrist, <a href="/A342577/b342577.txt">Table of n, a(n) for n = 0..5000</a>
%H Rémy Sigrist, <a href="/A342577/a342577.png">Illustration of initial terms</a>
%H Rémy Sigrist, <a href="/A342577/a342577_1.png">Colored representation of the tiles at distance <= 512</a> (where the color is function of the distance)
%H Rémy Sigrist, <a href="/A342577/a342577.txt">C# program for A342577</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. A342425, A342597.
%K nonn
%O 0,2
%A _Rémy Sigrist_, Mar 15 2021