A275198 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 14.
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, 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, 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
Offset: 0
Examples
Triangle begins:
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, 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,
...
Links
Crossrefs
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).
Programs
-
Mathematica
Mod[Flatten[Table[Binomial[n, k], {n, 0, 14}, {k, 0, n}]], 14] -
Python
from math import comb, isqrt from sympy.ntheory.modular import crt def A275198(n): w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1 d = int(not ~r & w) while True: r, a = divmod(r,7) w, b = divmod(w,7) c = c*comb(a,b)%7 if r<7 and w<7: c = c*comb(r,w)%7 break return crt([7,2],[c,d])[0] # Chai Wah Wu, May 01 2025