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A095141 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 6.

%I A095141 #33 Aug 04 2025 03:39:15
%S A095141 1,1,1,1,2,1,1,3,3,1,1,4,0,4,1,1,5,4,4,5,1,1,0,3,2,3,0,1,1,1,3,5,5,3,
%T A095141 1,1,1,2,4,2,4,2,4,2,1,1,3,0,0,0,0,0,0,3,1,1,4,3,0,0,0,0,0,3,4,1,1,5,
%U A095141 1,3,0,0,0,0,3,1,5,1,1,0,0,4,3,0,0,0,3,4,0,0,1,1,1,0,4,1,3,0,0,3,1,4,0,1,1
%N A095141 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 6.
%H A095141 Bill Gosper, <a href="/A095141/a095141.png">Pastel-colored illustration of triangle</a>
%H A095141 Ilya Gutkovskiy, <a href="/A275198/a275198.pdf">Illustrations (triangle formed by reading Pascal's triangle mod m)</a>
%H A095141 Ivan Korec, <a href="http://actamath.savbb.sk/acta0405.shtml">Definability of Pascal's Triangles Modulo 4 and 6 and Some Other Binary Operations from Their Associated Equivalence Relations</a>, Acta Univ. M. Belii Ser. Math. 4 (1996), pp. 53-66.
%H A095141 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A095141 T(i, j) = binomial(i, j) mod 6.
%t A095141 Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 6]
%t A095141 Graphics[Table[{%[Mod[Binomial[n, k], 6]/5], RegularPolygon[{4√3 (k - n/2), -6 n}, {4,π/6}, 6]}, {n, 0, 105}, {k, 0, n}]] (* Mma code for illustration, _Bill Gosper_, Aug 05 2017 *)
%o A095141 (Python)
%o A095141 from math import isqrt, comb
%o A095141 from sympy.ntheory.modular import crt
%o A095141 def A095141(n):
%o A095141     w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1
%o A095141     d = int(not ~r & w)
%o A095141     while True:
%o A095141         r, a = divmod(r,3)
%o A095141         w, b = divmod(w,3)
%o A095141         c = c*comb(a,b)%3
%o A095141         if r<3 and w<3:
%o A095141             c = c*comb(r,w)%3
%o A095141             break
%o A095141     return crt([3,2],[c,d])[0] # _Chai Wah Wu_, May 01 2025
%Y A095141 Cf. A007318, A047999, A083093, A034931, A095140, A095142, A034930, A095143, A008975, A095144, A095145, A034932.
%Y A095141 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), (this sequence) (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).
%K A095141 easy,nonn,tabl
%O A095141 0,5
%A A095141 _Robert G. Wilson v_, May 29 2004