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A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.

%I A008975 #41 May 01 2025 15:40:47
%S A008975 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,0,0,5,1,1,6,5,0,5,6,1,1,7,1,5,5,1,
%T A008975 7,1,1,8,8,6,0,6,8,8,1,1,9,6,4,6,6,4,6,9,1,1,0,5,0,0,2,0,0,5,0,1,1,1,
%U A008975 5,5,0,2,2,0,5,5,1,1,1,2,6,0,5,2,4,2,5,0,6,2,1,1,3,8,6,5,7,6,6
%N A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.
%H A008975 Reinhard Zumkeller, <a href="/A008975/b008975.txt">Rows n=0..150 of triangle, flattened</a>
%H A008975 Ilya Gutkovskiy, <a href="/A275198/a275198.pdf">Illustrations (triangle formed by reading Pascal's triangle mod m)</a>
%H A008975 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A008975 T(i, j) = binomial(i, j) mod 10.
%t A008975 Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 10] (* _Robert G. Wilson v_, May 26 2004 *)
%o A008975 (Haskell)
%o A008975 a008975 n k = a008975_tabl !! n !! k
%o A008975 a008975_row n = a008975_tabl !! n
%o A008975 a008975_tabl = iterate
%o A008975    (\row -> map (`mod` 10) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
%o A008975 -- _Reinhard Zumkeller_, Feb 24 2012
%o A008975 (Python)
%o A008975 from math import isqrt, comb
%o A008975 from sympy.ntheory.modular import crt
%o A008975 def A008975(n):
%o A008975     w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1
%o A008975     d = int(not ~r & w)
%o A008975     while True:
%o A008975         r, a = divmod(r,5)
%o A008975         w, b = divmod(w,5)
%o A008975         c = c*comb(a,b)%5
%o A008975         if r<5 and w<5:
%o A008975             c = c*comb(r,w)%5
%o A008975             break
%o A008975     return crt([5,2],[c,d])[0] # _Chai Wah Wu_, May 01 2025
%Y A008975 Cf. A007318, A047999, A083093, A034931, A034930, A034932.
%Y A008975 Cf. A208278 (row sums), A208279 (central terms), A208134 (number of zeros per row), A208280 (distinct terms per row).
%Y A008975 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), (this sequence) (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).
%K A008975 nonn,tabl,base
%O A008975 0,5
%A A008975 _N. J. A. Sloane_