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

0,5

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

Formula

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

Mathematica

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

Prog

(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
(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

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).

Sequence in context: A180181 A128629 A107065 * A140280 A140586 A339379

Adjacent sequences: A008972 A008973 A008974 * A008976 A008977 A008978

Keyword

nonn,tabl,base

Author

N. J. A. Sloane

Status

approved