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A095143 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 9.

%I A095143 #31 Jul 23 2025 00:57:11
%S A095143 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,1,1,5,1,1,6,6,2,6,6,1,1,7,3,8,8,3,
%T A095143 7,1,1,8,1,2,7,2,1,8,1,1,0,0,3,0,0,3,0,0,1,1,1,0,3,3,0,3,3,0,1,1,1,2,
%U A095143 1,3,6,3,3,6,3,1,2,1,1,3,3,4,0,0,6,0,0,4,3,3,1,1,4,6,7,4,0,6,6,0,4,7,6,4,1
%N A095143 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 9.
%H A095143 Ilya Gutkovskiy, <a href="/A275198/a275198.pdf">Illustrations (triangle formed by reading Pascal's triangle mod m)</a>
%H A095143 James G. Huard, Blair K. Spearman and Kenneth S. Williams, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa78/aa7843.pdf">Pascal's triangle (mod 9)</a>, Acta Arithmetica (1997), Volume: 78, Issue: 4, page 331-349.
%H A095143 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A095143 T(i, j) = binomial(i, j) mod 9.
%e A095143 Triangle begins:
%e A095143               1;
%e A095143             1,  1;
%e A095143           1,  2,  1;
%e A095143         1,  3,  3,  1;
%e A095143       1,  4,  6,  4,  1;
%e A095143     1,  5,  1,  1,  5,  1;
%e A095143   1,  6,  6,  2,  6,  6,  1;
%e A095143   ...
%t A095143 Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 9]
%o A095143 (Python)
%o A095143 from math import isqrt, comb
%o A095143 from gmpy2 import digits
%o A095143 def A095143(n):
%o A095143     g = (m:=isqrt(f:=n+1<<1))-(f<=m*(m+1))
%o A095143     k = n-comb(g+1,2)
%o A095143     if sum(int(d) for d in digits(k,3))+sum(int(d) for d in digits(g-k,3))-sum(int(d) for d in digits(g,3))>2: return 0
%o A095143     s, c, d = digits(g,3), 1, 0
%o A095143     w = (digits(k,3)).zfill(l:=len(s))
%o A095143     if l == 1: return comb(g,k)%9
%o A095143     for i in range(0,l-1):
%o A095143         r, t = s[i:i+2], w[i:i+2]
%o A095143         if (x:=int(r,3)) < (y:=int(t,3)):
%o A095143             d += (t[0]>r[0])+(t[1]>r[1])
%o A095143             if r[1]>=t[1]:
%o A095143                 c = c*comb(int(r[1],3),int(t[1],3))%9
%o A095143         else:
%o A095143             c = c*comb(x,y)%9
%o A095143     for i in range(1,l-1):
%o A095143         if w[i]>s[i] or (z:=comb(int(s[i],3),int(w[i],3))) == 3:
%o A095143             d -= 1
%o A095143         else:
%o A095143             c = c*pow(z,-1,9)%9
%o A095143     return c*3**d%9 # _Chai Wah Wu_, Jul 19 2025
%Y A095143 Cf. A007318, A047999, A083093, A034931, A095140, A095141, A095142, A034930, A008975, A095144, A095145, A034932.
%Y A095143 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), (this sequence) (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).
%K A095143 easy,nonn,tabl
%O A095143 0,5
%A A095143 _Robert G. Wilson v_, May 29 2004