A095141 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 6.

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

Offset

0,5

Links

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

Bill Gosper, Pastel-colored illustration of triangle

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

Ivan Korec, Definability of Pascal's Triangles Modulo 4 and 6 and Some Other Binary Operations from Their Associated Equivalence Relations, Acta Univ. M. Belii Ser. Math. 4 (1996), pp. 53-66.

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(i, j) = binomial(i, j) mod 6.

Mathematica

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 6]
Graphics[Table[{%[Mod[Binomial[n, k], 6]/5], RegularPolygon[{4√3 (k - n/2), -6 n}, {4, π/6}, 6]}, {n, 0, 105}, {k, 0, n}]] (* Mma code for illustration, Bill Gosper, Aug 05 2017 *)

Prog

(Python)
from math import isqrt, comb
from sympy.ntheory.modular import crt
def A095141(n):
    w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1
    d = int(not ~r & w)
    while True:
        r, a = divmod(r, 3)
        w, b = divmod(w, 3)
        c = c*comb(a, b)%3
        if r<3 and w<3:
            c = c*comb(r, w)%3
            break
    return crt([3, 2], [c, d])[0] # Chai Wah Wu, May 01 2025

Crossrefs

Cf. A007318, A047999, A083093, A034931, A095140, 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), A095140 (m = 5), (this sequence) (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).

Sequence in context: A027948 A360335 A370399 * A177974 A095140 A225043

Adjacent sequences: A095138 A095139 A095140 * A095142 A095143 A095144

Keyword

easy,nonn,tabl

Author

Robert G. Wilson v, May 29 2004

Status

approved