A095143 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 9.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 1, 1, 5, 1, 1, 6, 6, 2, 6, 6, 1, 1, 7, 3, 8, 8, 3, 7, 1, 1, 8, 1, 2, 7, 2, 1, 8, 1, 1, 0, 0, 3, 0, 0, 3, 0, 0, 1, 1, 1, 0, 3, 3, 0, 3, 3, 0, 1, 1, 1, 2, 1, 3, 6, 3, 3, 6, 3, 1, 2, 1, 1, 3, 3, 4, 0, 0, 6, 0, 0, 4, 3, 3, 1, 1, 4, 6, 7, 4, 0, 6, 6, 0, 4, 7, 6, 4, 1

Offset: 0

Author

Robert G. Wilson v, May 29 2004

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
              1;
            1,  1;
          1,  2,  1;
        1,  3,  3,  1;
      1,  4,  6,  4,  1;
    1,  5,  1,  1,  5,  1;
  1,  6,  6,  2,  6,  6,  1;
  ...
		

Links

• Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
• James G. Huard, Blair K. Spearman and Kenneth S. Williams, Pascal's triangle (mod 9), Acta Arithmetica (1997), Volume: 78, Issue: 4, page 331-349.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A047999, A083093, A034931, A095140, A095141, A095142, A034930, A008975, A095144, A095145, A034932.

Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930 (m = 8), (this sequence) (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

Programs

Mathematica

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 9]

Python

from math import isqrt, comb
from gmpy2 import digits
def A095143(n):
    g = (m:=isqrt(f:=n+1<<1))-(f<=m*(m+1))
    k = n-comb(g+1,2)
    if sum(int(d) for d in digits(k,3))+sum(int(d) for d in digits(g-k,3))-sum(int(d) for d in digits(g,3))>2: return 0
    s, c, d = digits(g,3), 1, 0
    w = (digits(k,3)).zfill(l:=len(s))
    if l == 1: return comb(g,k)%9
    for i in range(0,l-1):
        r, t = s[i:i+2], w[i:i+2]
        if (x:=int(r,3)) < (y:=int(t,3)):
            d += (t[0]>r[0])+(t[1]>r[1])
            if r[1]>=t[1]:
                c = c*comb(int(r[1],3),int(t[1],3))%9
        else:
            c = c*comb(x,y)%9
    for i in range(1,l-1):
        if w[i]>s[i] or (z:=comb(int(s[i],3),int(w[i],3))) == 3:
            d -= 1
        else:
            c = c*pow(z,-1,9)%9
    return c*3**d%9 # Chai Wah Wu, Jul 19 2025

Formula

T(i, j) = binomial(i, j) mod 9.