(50g) Delannoy numbers
06-02-2019, 11:34 PM (This post was last modified: 06-02-2019 11:38 PM by John Keith.)
Post: #1
 John Keith Senior Member Posts: 618 Joined: Dec 2013
(50g) Delannoy numbers
Delannoy numbers have many applications in combinatorics and number theory. Fortunately they are fast and easy to compute. Further details here.

The following program returns a rectangular array of Delannoy numbers. Given two integers referred to as m and n in the linked article, the program will return an array of m columns and n rows. The program actually returns a list of lists, and may be followed by AXL if a matrix is desired. The program requires the GoferLists library.

Code:
 \<< \-> m n   \<< 1 m NDUPN \->LIST 2 n     START DUP 2.       \<< +       \>> DOSUBS 1       \<< +       \>> Scanl     NEXT n \->LIST   \>> \>>

The next program returns the Delannoy triangle, also known as the tribonacci triangle. The numbers are the same as in the array but arranged in a triangle. Given an integer n, the program returns rows 0 through n of the triangle as a list of lists.

Code:
 \<< \-> n   \<< { 1 } DUP 1 + 2 n     START DUP2 2.       \<< +       \>> DOSUBS ADD 1 + 1 SWAP +     NEXT n 1 + \->LIST   \>> \>>

Finally, a program that computes the central Delannoy numbers which are the central column of the triangle and the main diagonal of the array. Given an integer n the program returns terms 0 through n of the sequence.

Code:
 \<< \-> n   \<< 1 3 2 n     FOR k DUP2 k 6 * 3 - * SWAP k 1 - * - k /     NEXT n 1 + \->LIST   \>> \>>

The last two programs will run on the HP-48G but are of limited usefulness since the numbers involved grow rapidly beyond 12 digits.
06-03-2019, 06:52 PM
Post: #2
 Luigi Vampa Member Posts: 246 Joined: Dec 2015
RE: (50g) Delannoy numbers
I never heard of Delannoy numbers. Thanks for sharing John.

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