A122881 Triangle read by rows: number of Catalan paths of 2n steps of all values less than or equal to m.

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, 1, 2, 5, 14, 42, 132, 429, 1341, 1597, 1, 2, 5, 14, 42, 132, 429, 1429, 4334, 4181

Offset

1,3

Comments

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 + 2*cos(2*Pi/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.

Links

Table of n, a(n) for n=1..55.

Formula

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.

Example

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;
  ...

Crossrefs

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

Sequence in context: A171840 A132309 A144224 * A210217 A334165 A370887

Adjacent sequences: A122878 A122879 A122880 * A122882 A122883 A122884

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 16 2006

Status

approved