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

%I A034930 #35 Jul 23 2025 00:57:45
%S A034930 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,
%T A034930 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,
%U A034930 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
%N A034930 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 8.
%H A034930 Reinhard Zumkeller, <a href="/A034930/b034930.txt">Rows n = 0..120 of triangle, flattened</a>
%H A034930 Ilya Gutkovskiy, <a href="/A275198/a275198.pdf">Illustrations (triangle formed by reading Pascal's triangle mod m)</a>
%H A034930 James G. Huard, Blair K. Spearman, and Kenneth S. Williams, <a href="https://doi.org/10.1006/eujc.1997.0146">Pascal's triangle (mod 8)</a>, European Journal of Combinatorics 19:1 (1998), pp. 45-62.
%H A034930 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A034930 T(n+1,k) = (T(n,k) + T(n,k-1)) mod 8. - _Reinhard Zumkeller_, Jul 12 2013
%t A034930 Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 8] (* _Robert G. Wilson v_, May 26 2004 *)
%o A034930 (Haskell)
%o A034930 a034930 n k = a034930_tabl !! n !! k
%o A034930 a034930_row n = a034930_tabl !! n
%o A034930 a034930_tabl = iterate
%o A034930    (\ws -> zipWith (\u v -> mod (u + v) 8) ([0] ++ ws) (ws ++ [0])) [1]
%o A034930 -- _Reinhard Zumkeller_, Jul 12 2013, Jun 21 2013
%o A034930 (Python)
%o A034930 from math import comb, isqrt
%o A034930 def A034930(n):
%o A034930     g = (m:=isqrt(f:=n+1<<1))-(f<=m*(m+1))
%o A034930     k = n-comb(g+1,2)
%o A034930     if k.bit_count()+(g-k).bit_count()-g.bit_count()>2: return 0
%o A034930     def g1(s,w,e):
%o A034930         c, d = 1, 0
%o A034930         if len(s) == 0: return c, d
%o A034930         a, b = int(s,2), int(w,2)
%o A034930         if a>=b:
%o A034930             k = comb(a,b)&7
%o A034930             j = (~k & k-1).bit_length()
%o A034930             d += j*e
%o A034930             k >>= j
%o A034930             c = c*pow(k,e,8)&7
%o A034930         else:
%o A034930             if int(s[0:1],2)<int(w[0:1],2): d += e
%o A034930             c0, d0 = g1(s[1:],w[1:],e)
%o A034930             c = c*c0&7
%o A034930             d += d0
%o A034930         return c, d
%o A034930     s = bin(g)[2:].zfill(3)
%o A034930     w = bin(k)[2:].zfill(l:=len(s))
%o A034930     c, d = g1(s[:3],w[:3],1)
%o A034930     for i in range(1,l-1):
%o A034930         c0, d0 = g1(s[i:i+3],w[i:i+3],1)
%o A034930         c1, d1 = g1(s[i:i+2],w[i:i+2],-1)
%o A034930         c = c*c0*c1&7
%o A034930         d += d0+d1
%o A034930     return (c<<d)&7 # _Chai Wah Wu_, Jul 20 2025
%Y A034930 Cf. A007318, A047999, A083093, A034931, A008975, A034932.
%Y A034930 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).
%K A034930 nonn,tabl
%O A034930 0,5
%A A034930 _N. J. A. Sloane_