A275198 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..119.

Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

T(n, k) = binomial(n, k) mod 14.

a(n) = A070696(A007318(n)).

EXAMPLE

Triangle begins:

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,

...

MATHEMATICA

Mod[Flatten[Table[Binomial[n, k], {n, 0, 14}, {k, 0, n}]], 14]

PROG

(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

CROSSREFS

Cf. A007318, A070696.

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).

Sequence in context: A214846 A061676 A180182 * A095145 A095144 A339359

Adjacent sequences: A275195 A275196 A275197 * A275199 A275200 A275201

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Aug 11 2016

STATUS

approved