This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349221 #69 Feb 06 2025 22:01:13
%S A349221 1,1,0,1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,
%T A349221 1,0,1,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,
%U A349221 0,0,1,1,1,0,0,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0
%N A349221 Triangle read by rows: T(n, k) = 1 if k divides binomial(n-1, k-1), T(n, k) = 0 otherwise (n >= 1, 1 <= k <= n).
%C A349221 Similar to A054521 as gcd(n, k) = 1 => k divides binomial(n-1, k-1) but not equivalent as the converse is not true, the earliest example being T(10,4) where 4 divides binomial(9,3) = 84 but gcd(10,4) is not 1. Question: What characterizes the cases where this triangle differs from A054521?
%C A349221 The period of the k-th column is given by A349593(k-1, k) = k * Product_{prime p|k} p^(floor(log(k-1)/log(p))). - _Jianing Song_, Nov 29 2021
%C A349221 {T(n, k)} is the sum of triangles [k|binomial(n-1, k-1) AND gcd(n, k) = j], n >= 1, 1 <= k <= n, j >= 1, where [] is the Iverson bracket. For j > 1, bitmaps of these triangles suggest simpler fractal gaskets that combine to produce the "shadowing" effect observed in the bitmap of {T(n, k)} provided in the LINKS section. For prime j, the bitmaps suggest a fractal (Hausdorff) dimension of log(A000217(j))/log(j) = log(j(j + 1)/2)/log(j), which is the same as that of the gasket formed by taking the Pascal triangle (A007318) mod j (see Bondarenko reference). - _Richard L. Ollerton_, Dec 10 2021
%D A349221 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%H A349221 Michael De Vlieger, <a href="/A349221/b349221.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)
%H A349221 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
%H A349221 Michael De Vlieger, <a href="/A349221/a349221.png">Bitmap</a> of rows 1 <= n <= 2^10, showing 1 as black and 0 as white.
%H A349221 Michael De Vlieger, <a href="/A349221/a349221.txt">Table of b(n)</a> for n = 1..3322, where b(n) is the compactification of row n of a(n) as a binary number.
%H A349221 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A349221 T(n, k) = [k|binomial(n-1, k-1)] = Sum_{j>=1} [k|binomial(n-1, k-1) AND gcd(n, k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. (The j = 1 case is A054521.)
%F A349221 T(n, k) = T(n, n-k), n > 1, 1 <= k < n.
%e A349221 The triangle T(n, k) begins:
%e A349221 n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e A349221 1: 1
%e A349221 2: 1 0
%e A349221 3: 1 1 0
%e A349221 4: 1 0 1 0
%e A349221 5: 1 1 1 1 0
%e A349221 6: 1 0 0 0 1 0
%e A349221 7: 1 1 1 1 1 1 0
%e A349221 8: 1 0 1 0 1 0 1 0
%e A349221 9: 1 1 0 1 1 0 1 1 0
%e A349221 10: 1 0 1 1 0 1 1 0 1 0
%e A349221 11: 1 1 1 1 1 1 1 1 1 1 0
%e A349221 12: 1 0 0 0 1 1 1 0 0 0 1 0
%e A349221 13: 1 1 1 1 1 1 1 1 1 1 1 1 0
%e A349221 14: 1 0 1 0 1 0 0 0 1 0 1 0 1 0
%e A349221 15: 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
%e A349221 ...
%e A349221 Differences between this example and that for A054521 occur at (n,k) = (10,4), (10,6), and (12,6).
%t A349221 Table[Boole[Mod[Binomial[n - 1, k - 1], k] == 0], {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Nov 11 2021 *)
%o A349221 (PARI) row(n) = vector(n, k, !(binomial(n-1,k-1) % k)); \\ _Michel Marcus_, Nov 11 2021
%Y A349221 Cf. A007318, A051731, A054521.
%K A349221 nonn,tabl,look
%O A349221 1,1
%A A349221 _Richard L. Ollerton_, Nov 11 2021