motzkin_numbers.sf

#!/usr/bin/ruby

# Generate the first `n` Motzkin numbers (the numbers on the hypotenuse of the Motzkin triangle).

# Motzkin's triangle:
#   1
#   1    1
#   1    2    2
#   1    3    5    4
#   1    4    9   12    9
#   1    5   14   25   30   21
#   1    6   20   44   69   76   51
#   1    7   27   70  133  189  196  127
#   1    8   35  104  230  392  518  512  323
#   1    9   44  147  369  726 1140 1422 1353  835
#   1   10   54  200  560 1242 2235 3288 3915 3610 2188
#   1   11   65  264  814 2002 4037 6765 9438 10813 9713 5798

# There is no Motzkin number divisible by 8.

# Eu Sen-Peng & Liu Shu-Chung & Yeh, Yeong-nan proved in 2008 that no Motzkin number
# is a multiple of 8, in their paper "Catalan and Motzkin numbers modulo 4 and 8".
# https://www.math.sinica.edu.tw/www/file_upload/mayeh/2008CatalanandMotzkinnumbersmodulo4and8.pdf

# See also:
#   https://oeis.org/A001006
#   https://en.wikipedia.org/wiki/Motzkin_number
#   https://oeis.org/wiki/User:Peter_Luschny/MotzkinNumbers

# Traslation of the Sage program by Peter Luschny from https://oeis.org/A001006

func motzkin_numbers(N) {

    var (a, b, n) = (0, 1, 1)

    N.of {
        var M = b//n        #/
        n += 1
        (a, b) = (b, (3*(n-1)*n*a + (2*n - 1)*n*b) // ((n+1)*(n-1)))        #/
        M
    }
}

say motzkin_numbers(30)

__END__
1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829