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%I #15 Jan 09 2024 16:34:56
%S 1,1,1,3,1,4,1,7,11,1,8,15,1,11,38,41,1,12,46,56,1,15,81,186,153,1,16,
%T 93,232,209,1,19,140,49,859,571,1,20,156,592,1091,780,1,23,215,1044,
%U 2774,3821,2131,1,24,235,1200,3366,4912,2911,1,27,306,1885,6810,14418
%N Irregular triangle by rows derived from variants of Cartan matrices: 1's in the super and subdiagonals and 3,4,4,4,... in the main diagonal alternating with 4,4,4,...
%C Row sums starting with row 2 = A136211: (1, 4, 5, 19, 24, ...) = denominators in convergents to [1, 3, 1, 3, 1, 3, ...].
%C Rightmost terms in each row = A002530, denominators in convergents to [1, 2, 1, 2, 1, 2, ...], prefaced with a 1 for row 1. Odd-indexed row rightmost terms = Product_{k=1..(n-1)/2} (2 + 4*cos^2(k*2*Pi/n))
%C Example: x^3 - 11x^2 + 38x + 41 = row 7 relating to the heptagon, with roots = 5.246979..., 3.554958..., and 2.19806226, product = 41 (same result as using the product formula).
%C Even-indexed rows related to even-sided regular polygons; but use the product formula: rightmost terms in even rows >2 = Product_{k=1..(n-2)/2} (2 + 4*cos^2(k*Pi/n)).
%C Using the product formula or root products with row 8 relating to the octagon, we obtain 5.414..., * 4 * 2.585... = 56, rightmost term of row 8.
%C Shifted columns of A180062 = triangle A180063.
%F Triangle read by rows generated from Cartan-like matrices, 1's in the super and subdiagonals, with alternates of (3,4,4,4,...) for odd-indexed rows and (4,4,4,...) for even-indexed rows. The first nontrivial matrix = [3,1; 1,4] with charpoly x^2 - 7x + 11, becoming row 5: (1, 7, 11); generating row 3: (x^2 - 7x + 11). Rows begin 1; 1; 1,3; 1,4;...
%F The first few rows can be constructed using the following set of rules:
%F Rightmost terms in each row = A002530, denominators in continued fraction [1, 2, 1, 2, 1, 2,...] = (1, 3, 4, 11, 15,...), while row sums = A136211, denominators in [1, 3, 1, 3, 1, 3,...] = (1, 4, 5, 19, 24,...) given row 1 = 1.
%F Negative signs in the charpolys are changed to + in the triangle.
%e First few rows of the triangle:
%e 1;
%e 1;
%e 1, 3;
%e 1, 4;
%e 1, 7, 11;
%e 1, 8, 15;
%e 1, 11, 38, 41;
%e 1, 12, 46, 56;
%e 1, 15, 81, 186, 153;
%e 1, 16, 93, 232, 209;
%e 1, 19, 140, 499, 859, 571;
%e 1, 20, 156, 592, 1091, 780;
%e 1, 23, 215, 1044, 2774, 3821, 2131;
%e 1, 24, 235, 1200, 3366, 4912, 2911;
%e 1, 27, 306, 1885, 6810, 14418, 26556, 7953;
%e 1, 28, 330, 2120, 8010, 17784, 21468, 10864;
%e 1, 31, 413, 3086, 14135, 40614, 71454, 70356, 29681;
%e 1, 32, 441, 3416, 16255, 48624, 89238, 91824, 40545;
%e 1, 35, 536, 4711, 26173, 95269, 227100, 341754, 294549, 110771;
%e 1, 36, 568, 5152, 29589, 111524, 275724, 430992, 386373, 151316;
%e ...
%e Examples:
%e Row 7 = x^3 - 11 x^2 + 38x + 41, charpoly of the 3 X 3 matrix [3,1,0; 1,4,1; 0,1,4], then changing (-) signs to (+).
%e Row 8 = x^3 - 12x^2 + 46x - 56, = charpoly of [4,1,0; 1,4,1; 0,1,4].
%Y Cf. A002530, A136211, A180063, A180063.
%K nonn,tabf
%O 1,4
%A _Gary W. Adamson_, Aug 08 2010
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