A095142 - OEIS

A095142

Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 7.

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

{T(n,k)} is a fractal gasket with fractal (Hausdorff) dimension log(A000217(7))/log(7) = log(28)/log(7) = 1.71241... (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.

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

LINKS

Table of n, a(n) for n=0..104.

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.

Ashley Melia 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

FORMULA

T(i, j) = binomial(i, j) mod 7.

MATHEMATICA

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 7]

PROG

(Python)

from math import comb, isqrt

def A095142(n):

def f(m, k):

if m<7 and k<7: return comb(m, k)%7

c, a = divmod(m, 7)

d, b = divmod(k, 7)

return f(c, d)*f(a, b)%7

return f(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)), n-comb(r+1, 2)) # Chai Wah Wu, Apr 30 2025

CROSSREFS

Cf. A007318, A047999, A083093, A034931, A095140, A095141, 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), A095140 (m = 5), A095141 (m = 6), (this sequence) (m = 7), A034930 (m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

Sequence in context: A306309 A123264 A034930 * A180171 A140822 A381451

Adjacent sequences: A095139 A095140 A095141 * A095143 A095144 A095145

KEYWORD

easy,nonn,tabl

AUTHOR

Robert G. Wilson v, May 29 2004

STATUS

approved