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%I #21 Nov 30 2019 10:42:01
%S 1,1,2,1,4,2,1,4,5,2,1,4,8,4,2,1,4,8,8,4,2,1,4,8,12,6,4,2,1,4,8,12,11,
%T 6,4,2,1,4,8,12,16,8,6,4,2,1,4,8,12,16,14,8,6,4,2,1,4,8,12,16,20,10,8,
%U 6,4,2,1,4,8,12,16,20,17,10,8,6,4,2
%N Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n squares cut from the square grid by cuts at 45 degrees to grid lines.
%C By the "width" of the strip is meant the number of squares in a corner-to-corner ring around the cylinder.
%C For the case when the cuts are parallel to grid lines, see A329501.
%C See A329508 ... for coordination sequences for cylinders formed by rolling up the hexagonal grid ("carbon nanotubes").
%H N. J. A. Sloane, <a href="/A329504/a329504_1.pdf">Illustration for rows 1 through 5</a>, showing vertices of cylinder labeled with distance from base point (c = n is the width (or circumference)). The cylinders are formed by identifying the black lines.
%H <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a>
%F Let theta = (1+x)/(1-x). The g.f. for the coordination sequence for row n is theta*(1+2x+2x^2+...+2x^(n-1)-(n-1)*x^n).
%e Array begins:
%e 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 1, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
%e 1, 4, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, ...
%e 1, 4, 8, 12, 11, 8, 8, 8, 8, 8, 8, 8, ...
%e 1, 4, 8, 12, 16, 14, 10, 10, 10, 10, 10, 10, ...
%e 1, 4, 8, 12, 16, 20, 17, 12, 12, 12, 12, 12, ...
%e 1, 4, 8, 12, 16, 20, 24, 20, 14, 14, 14, 14, ...
%e 1, 4, 8, 12, 16, 20, 24, 28, 23, 16, 16, 16, ...
%e 1, 4, 8, 12, 16, 20, 24, 28, 32, 26, 18, 18, ...
%e 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 29, 20, ...
%e ...
%e The initial antidiagonals are:
%e 1,
%e 1,2,
%e 1,4,2,
%e 1,4,5,2,
%e 1,4,8,4,2,
%e 1,4,8,8,4,2,
%e 1,4,8,12,6,4,2,
%e 1,4,8,12,11,6,4,2,
%e 1,4,8,12,16,8,6,4,2,
%e ...
%Y Rows 2,3,4 are A329505, A329506, A329507.
%Y Cf. A008574, A329501-A329517.
%K nonn,tabl,easy
%O 1,3
%A _N. J. A. Sloane_, Nov 19 2019
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