(2,1)-Pascal triangle

Rows zero to five of (2,1)-Pascal triangle

In mathematics, the (2,1)-Pascal triangle (mirrored Lucas triangle[1])is a triangular array.

The rows of the (2,1)-Pascal triangle (sequence A029653 in the OEIS)[2] are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.

The triangle is based on the Pascal's Triangle with the second line being (2,1) and the first cell of each row set to 2.

This construction is related to the binomial coefficients by Pascal's rule, with one of the terms being .

Patterns and properties[]

(2,1)-Pascal triangle has many properties and contains many patterns of numbers. It can be seen as a sister of the Pascal's triangle, in the same way that a Lucas sequence is a sister sequence of the Fibonacci sequence.[citation needed]

Rows[]

Diagonals[]

The diagonals of Pascal's triangle contain the figurate numbers of simplices:

Overall patterns and properties[]

Sierpinski triangle
(2,1)-Pascal triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward movements are considered.
1
2 1
2 3 1
2 5 4 1
2 7 9 5 1
2 9 16 14 6 1
2 11 25 30 20 7 1
2 13 36 55 50 27 8 1
2 15 49 91 105 77 35 9 1
1
2 1
2 3 1
2 5 4 1
2 7 9 5 1
2 9 16 14 6 1
2 11 25 30 20 7 1
2 13 36 55 50 27 8 1
2 15 49 91 105 77 35 9 1

References[]

  1. ^ "(1,2)-Pascal triangle - OeisWiki". oeis.org. Retrieved 2016-02-23.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A029653 (Numbers in (2,1)-Pascal triangle (by row))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2015-12-24.
  3. ^ Wolfram, S. (1984). "Computation Theory of Cellular Automata". Comm. Math. Phys. 96: 15–57. Bibcode:1984CMaPh..96...15W. doi:10.1007/BF01217347.
  4. ^ "An Exact Value For The Fine Structure Constant. - Page 7 - Physics and Mathematics". Science Forums. Retrieved 2016-02-01.