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

%I A382726 #23 Aug 15 2025 15:57:39
%S A382726 1,3,6,10,15,21,28,30,34,40,48,58,70,84,87,93,102,114,129,147,168,172,
%T A382726 180,192,208,228,252,280,285,295,310,330,355,385,420,426,438,456,480,
%U A382726 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
%N A382726 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 7.
%C A382726 Partial sums of A382720. - _James C. McMahon_, Aug 15 2025
%H A382726 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2408.06817">Periodic minimum in the count of binomial coefficients not divisible by a prime</a>, arXiv:2408.06817 [math.NT], 2024.
%t A382726 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 *)
%o A382726 (Python)
%o A382726 from math import prod
%o A382726 from gmpy2 import digits
%o A382726 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
%o A382726 (Python)
%o A382726 from math import prod
%o A382726 from gmpy2 import digits
%o A382726 def A382726(n):
%o A382726     d = list(map(lambda x:int(x)+1,digits(n+1,7)[::-1]))
%o A382726     return sum((b-1)*prod(d[a:])*28**a for a, b in enumerate(d))>>1 # _Chai Wah Wu_, Aug 13 2025
%Y A382726 Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382731.
%K A382726 nonn
%O A382726 0,2
%A A382726 _N. J. A. Sloane_, Apr 23 2025