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A180062 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,... 2
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, 93, 232, 209, 1, 19, 140, 49, 859, 571, 1, 20, 156, 592, 1091, 780, 1, 23, 215, 1044, 2774, 3821, 2131, 1, 24, 235, 1200, 3366, 4912, 2911, 1, 27, 306, 1885, 6810, 14418 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums starting with row 2 = A136211: (1, 4, 5, 19, 24, ...) = denominators in convergents to [1, 3, 1, 3, 1, 3, ...].

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))

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).

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)).

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.

Shifted columns of A180062 = triangle A180063.

LINKS

Table of n, a(n) for n=1..62.

FORMULA

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;...

The first few rows can be constructed using the following set of rules:

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.

Negative signs in the charpolys are changed to + in the triangle.

EXAMPLE

First few rows of the triangle:

  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,  93,  232,   209;

  1, 19, 140,  499,   859,    571;

  1, 20, 156,  592,  1091,    780;

  1, 23, 215, 1044,  2774,   3821,   2131;

  1, 24, 235, 1200,  3366,   4912,   2911;

  1, 27, 306, 1885,  6810,  14418,  26556,   7953;

  1, 28, 330, 2120,  8010,  17784,  21468,  10864;

  1, 31, 413, 3086, 14135,  40614,  71454,  70356,  29681;

  1, 32, 441, 3416, 16255,  48624,  89238,  91824,  40545;

  1, 35, 536, 4711, 26173,  95269, 227100, 341754, 294549, 110771;

  1, 36, 568, 5152, 29589, 111524, 275724, 430992, 386373, 151316;

  ...

Examples:

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 (+).

Row 8 = x^3 - 12x^2 + 46x - 56, = charpoly of [4,1,0; 1,4,1; 0,1,4].

CROSSREFS

Cf. A002530, A136211, A180063, A180063.

Sequence in context: A273133 A318841 A300238 * A079546 A014413 A262072

Adjacent sequences:  A180059 A180060 A180061 * A180063 A180064 A180065

KEYWORD

nonn,tabf

AUTHOR

Gary W. Adamson, Aug 08 2010

STATUS

approved

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Last modified May 28 04:51 EDT 2020. Contains 334671 sequences. (Running on oeis4.)