A034930 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 8.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
• James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 8), European Journal of Combinatorics 19:1 (1998), pp. 45-62.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

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

Programs

Haskell

a034930 n k = a034930_tabl !! n !! k
a034930_row n = a034930_tabl !! n
a034930_tabl = iterate
   (\ws -> zipWith (\u v -> mod (u + v) 8) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Jul 12 2013, Jun 21 2013

Mathematica

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 8] (* Robert G. Wilson v, May 26 2004 *)

Python

from math import comb, isqrt
def A034930(n):
    g = (m:=isqrt(f:=n+1<<1))-(f<=m*(m+1))
    k = n-comb(g+1,2)
    if k.bit_count()+(g-k).bit_count()-g.bit_count()>2: return 0
    def g1(s,w,e):
        c, d = 1, 0
        if len(s) == 0: return c, d
        a, b = int(s,2), int(w,2)
        if a>=b:
            k = comb(a,b)&7
            j = (~k & k-1).bit_length()
            d += j*e
            k >>= j
            c = c*pow(k,e,8)&7
        else:
            if int(s[0:1],2)Chai Wah Wu, Jul 20 2025

Formula

T(n+1,k) = (T(n,k) + T(n,k-1)) mod 8. - Reinhard Zumkeller, Jul 12 2013