A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 0, 0, 5, 1, 1, 6, 5, 0, 5, 6, 1, 1, 7, 1, 5, 5, 1, 7, 1, 1, 8, 8, 6, 0, 6, 8, 8, 1, 1, 9, 6, 4, 6, 6, 4, 6, 9, 1, 1, 0, 5, 0, 0, 2, 0, 0, 5, 0, 1, 1, 1, 5, 5, 0, 2, 2, 0, 5, 5, 1, 1, 1, 2, 6, 0, 5, 2, 4, 2, 5, 0, 6, 2, 1, 1, 3, 8, 6, 5, 7, 6, 6

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• base

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A047999, A083093, A034931, A034930, A034932.

Cf. A208278 (row sums), A208279 (central terms), A208134 (number of zeros per row), A208280 (distinct terms per row).

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), (this sequence) (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

Programs

Haskell

a008975 n k = a008975_tabl !! n !! k
a008975_row n = a008975_tabl !! n
a008975_tabl = iterate
   (\row -> map (`mod` 10) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Feb 24 2012

Mathematica

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 10] (* Robert G. Wilson v, May 26 2004 *)

Python

from math import isqrt, comb
from sympy.ntheory.modular import crt
def A008975(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,5)
        w, b = divmod(w,5)
        c = c*comb(a,b)%5
        if r<5 and w<5:
            c = c*comb(r,w)%5
            break
    return crt([5,2],[c,d])[0] # Chai Wah Wu, May 01 2025

Formula

T(i, j) = binomial(i, j) mod 10.