A095144 - OEIS

A095144

Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 11.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 4, 9, 4, 6, 1, 1, 7, 10, 2, 2, 10, 7, 1, 1, 8, 6, 1, 4, 1, 6, 8, 1, 1, 9, 3, 7, 5, 5, 7, 3, 9, 1, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 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(11))/log(11) = log(66)/log(11) = 1.74722... (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

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

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.

Zubeyir Cinkir and Aysegul Ozturkalan, An extension of Lucas Theorem, arXiv:2309.00109 [math.NT], 2023. See Figures 3 and 4 p. 6.

Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)

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 11.

From Robert Israel, Jan 02 2019: (Start)

T(n,k) = (T(n-1,k-1) + T(n-1,k)) mod 11 with T(n,0) = 1.

T(n,k) = (Product_i binomial(n_i, k_i)) mod 11, where n_i and k_i are the base-11 digits of n and k. (End)

MAPLE

R[0]:= 1:

for n from 1 to 20 do

R[n]:= op([R[n-1], 0] + [0, R[n-1]] mod 11);

od:

for n from 0 to 20 do R[n] od; # Robert Israel, Jan 02 2019

MATHEMATICA

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 11]

PROG

(Python)

from math import isqrt, comb

def A095144(n):

def f(m, k):

if m<11 and k<11: return comb(m, k)%11

c, a = divmod(m, 11)

d, b = divmod(k, 11)

return f(c, d)*f(a, b)%11

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, A095142, A034930, A095143, A008975, 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), A095143 (m = 9), A008975 (m = 10), (this sequence) (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

Sequence in context: A180182 A275198 A095145 * A339359 A144398 A034932

Adjacent sequences: A095141 A095142 A095143 * A095145 A095146 A095147

KEYWORD

easy,nonn,tabl

AUTHOR

Robert G. Wilson v, May 29 2004

STATUS

approved