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,...

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

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 * A340072 A079546 A014413

Adjacent sequences: A180059 A180060 A180061 * A180063 A180064 A180065

Keyword

nonn,tabf

Author

Gary W. Adamson, Aug 08 2010

Status

approved