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A275198 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 14.

%I A275198 #40 May 01 2025 15:40:51
%S A275198 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,1,6,1,6,1,1,7,7,7,7,
%T A275198 7,7,1,1,8,0,0,0,0,0,8,1,1,9,8,0,0,0,0,8,9,1,1,10,3,8,0,0,0,8,3,10,1,
%U A275198 1,11,13,11,8,0,0,8,11,13,11,1,1,12,10,10,5,8,0,8,5,10,10,12,1,1,13,8,6,1,13,8,8,13,1,6,8,13,1,1,0,7,0,7,0,7,2,7,0,7,0,7,0,1
%N A275198 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 14.
%H A275198 Ilya Gutkovskiy, <a href="/A275198/a275198.pdf">Illustrations (triangle formed by reading Pascal's triangle mod m)</a>
%H A275198 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A275198 T(n, k) = binomial(n, k) mod 14.
%F A275198 a(n) = A070696(A007318(n)).
%e A275198 Triangle begins:
%e A275198                       1,
%e A275198                     1,  1,
%e A275198                   1,  2,  1,
%e A275198                 1,  3,  3,  1,
%e A275198               1,  4,  6,  4,  1,
%e A275198             1,  5, 10, 10,  5,  1,
%e A275198           1,  6,  1,  6,  1,  6,  1,
%e A275198         1,  7,  7,  7,  7,  7,  7,  1,
%e A275198       1,  8,  0,  0,  0,  0,  0,  8,  1,
%e A275198     1,  9,  8,  0,  0,  0,  0,  8,  9,  1,
%e A275198   1, 10,  3,  8,  0,  0,  0,  8,  3, 10,  1,
%e A275198   ...
%t A275198 Mod[Flatten[Table[Binomial[n, k], {n, 0, 14}, {k, 0, n}]], 14]
%o A275198 (Python)
%o A275198 from math import comb, isqrt
%o A275198 from sympy.ntheory.modular import crt
%o A275198 def A275198(n):
%o A275198     w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1
%o A275198     d = int(not ~r & w)
%o A275198     while True:
%o A275198         r, a = divmod(r,7)
%o A275198         w, b = divmod(w,7)
%o A275198         c = c*comb(a,b)%7
%o A275198         if r<7 and w<7:
%o A275198             c = c*comb(r,w)%7
%o A275198             break
%o A275198     return crt([7,2],[c,d])[0] # _Chai Wah Wu_, May 01 2025
%Y A275198 Cf. A007318, A070696.
%Y A275198 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), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), (this sequence) (m = 14), A034932 (m = 16).
%K A275198 nonn,tabl
%O A275198 0,5
%A A275198 _Ilya Gutkovskiy_, Aug 11 2016