1,3

Convergents of k-th diagonals relate to (2k+3)-Polygons; e.g. right border relates to the Pentagon (N=5), next border relates to the Heptagon.(N=7). Convergents of the diagonals are 2 + 2Cos 2Pi/N and are roots to Morgan-Voyce polynomials. k2 diagonal = A080937, number of Catalan paths of 2n steps of all values less than or equal to 5. k3 diagonal = A080938, number of Catalan paths of 2n steps of all values less than or equal to 7.

Begin with polygonal matrices of the form (exemplified by the Heptagonal matrix M3: [1, 1, 1; 1, 1, 0; 1, 0, 0]). Let matrix P3 = 1 / M3^2; then for n X n matrices P2, P3, P4...perform P^n * [1, 0, 0] letting this vector = k-th diagonal of the triangle.

For the right border, odd indexed Fibonacci numbers (1, 2, 5, 13, 34...), we begin with (M2) = [1, 1; 1, 0], then P2 = [1, -1; -1, 2] = 1/(M2)^2. Performing (P2)^n * [1,0] we extract the left vector (1, 2, 5, 13...), making it the right border of the triangle, k1 diagonal.

For the next diagonal going to the left, we begin with the Heptagonal matrix M3 = [1, 1, 1; 1, 1, 0; 1, 0, 0], take the inverse square (P3) and then perform the analogous operation getting 1, 2, 5, 14, 42...

First few rows of the triangle are:

1;

1, 2;

1, 2, 5;

1, 2, 5, 13;

1, 2, 5, 14, 34;

1, 2, 5, 14, 42, 89;

1, 2, 5, 14, 42, 131, 233;

1, 2, 5, 14, 42, 132, 417, 610;

...

Cf. A112880, A001519, A000108, A080937, A080938.

Sequence in context: A002211 A132309 A144224 this_sequence A135506 A068822 A090079

Adjacent sequences: A122878 A122879 A122880 this_sequence A122882 A122883 A122884

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 16 2006