A382726 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 7.

1, 3, 6, 10, 15, 21, 28, 30, 34, 40, 48, 58, 70, 84, 87, 93, 102, 114, 129, 147, 168, 172, 180, 192, 208, 228, 252, 280, 285, 295, 310, 330, 355, 385, 420, 426, 438, 456, 480, 510, 546, 588, 595, 609, 630, 658, 693, 735, 784, 786, 790, 796, 804, 814, 826, 840, 844, 852, 864, 880, 900, 924, 952, 958, 970, 988, 1012, 1042, 1078

Offset: 0

Author

N. J. A. Sloane, Apr 23 2025

Keywords

• nonn

Comments

Partial sums of A382720. - James C. McMahon, Aug 15 2025

Links

• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024.

Crossrefs

Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382731.

Programs

Mathematica

a[n_]:=(n^2+3n+2)/2-Count[Mod[Flatten[Table[Binomial[m, k], {m, 0,n}, {k, 0,m}]] ,7],0];Array[a,69,0] (* James C. McMahon, Aug 15 2025 *)

Python

from math import prod
from gmpy2 import digits
def A382726(n): return sum(prod(int(d)+1 for d in digits(m,7)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

Python

from math import prod
from gmpy2 import digits
def A382726(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,7)[::-1]))
    return sum((b-1)*prod(d[a:])*28**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025