The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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