A095140 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 5.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 1, 4, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 3, 3, 1, 0, 1, 3, 3, 1, 1, 4, 1, 4, 1, 1, 4, 1, 4, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 2, 4, 2, 0, 0, 1, 2, 1, 1, 3, 3, 1, 0, 2, 1, 1, 2, 0, 1, 3, 3, 1

Offset: 0

Author

Robert G. Wilson v, May 29 2004

Keywords

• easy
• nonn
• tabl

Comments

{T(n,k)} is a fractal gasket with fractal (Hausdorff) dimension log(A000217(5))/log(5) = log(15)/log(5) = 1.68260... (see Reiter reference). Replacing values greater than 1 with 1 produces a binary gasket with the same dimension (see Bondarenko reference). - Richard L. Ollerton, Dec 14 2021

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

• Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
• A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A047999, A083093, A034931, A095141, A095142, A034930, A095143, 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), (this sequence) (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (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}]], 5]

Python

from math import isqrt, comb
def A095140(n):
    def f(m,k):
        if m<5 and k<5: return comb(m,k)%5
        c,a = divmod(m,5)
        d,b = divmod(k,5)
        return f(c,d)*f(a,b)%5
    return f(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2)) # Chai Wah Wu, Apr 30 2025

Formula

T(i, j) = binomial(i, j) mod 5.