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%I #28 Jan 24 2014 11:27:24
%S 1,4,11,24,25,47,48,49,50,80,82,83,84,85,86,87,88,89,90,91,137
%N Irregular triangle read by rows: Possible numbers of pieces resulting from cutting a convex n-sided polygon along all its diagonals.
%C Beginning from number of sides equal to 18 the terms no longer increase between rows. For example, the number of pieces for the regular 18-gon is fewer than the number of pieces for regular 17-gon.
%C Obviously there exists a number k_0 such that k_0 is not in the sequence and k is in the sequence for all k > k_0.
%H V.A. Letsko, M.A. Voronina <a href="http://grani.vspu.ru/files/publics/1301378772.pdf">Classification of convex polygons</a>, Grani Poznaniya, 1(11), 2011. (in Russian)
%H Vladimir Letsko, <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:problem_102">Mathematical Marathon at vspu, Problem 102</a> (in Russian)
%H Vladimir Letsko <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:illustrations_102_co">Illustration of all cases for number of sides from 3 to 8</a>
%F a(n,s_1,...,s_m) = A006522(n) - sum_{k=1}^m s_k*k*(k+1)/2, where m = floor(n/2)-2 and s_k denotes number of inner points in which exactly k+2 diagonals are intersected.
%e The beginning of the irregular triangle is:
%e 3| 1
%e 4| 4
%e 5| 11
%e 6| 24, 25
%e 7| 47, 48, 49, 50,
%e 8| 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91
%e 9| 137 (incomplete)
%Y Cf. A006522, A160860, A007678.
%K tabf,nonn
%O 3,2
%A _Vladimir Letsko_, Oct 11 2013
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