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 A290691 #33 Apr 07 2020 23:12:03
%S A290691 1,0,2,1,1,3,0,0,2,4,2,1,1,3,5,1,0,0,2,4,6,0,3,1,1,3,5,7,2,2,0,0,2,4,
%T A290691 6,8,1,1,4,1,1,3,5,7,9,0,0,3,0,0,2,4,6,8,10,3,2,2,5,1,1,3,5,7,9,11,2,
%U A290691 1,1,4,0,0,2,4,6,8,10,12,1,0,0,3,6,1,1
%N A290691 Triangle read by rows: infinite braid made of periodically-colored yarns in which the crossing of two adjacent yarns occurs when two color 0's are side by side (see comments).
%C A290691 Construction: row numbers start at n = 3; column numbers run from k = 1 to k = n - 2. For all y >= 2, a yarn called "yarn y" is made of the repeated (y - 1, y - 2, ..., 1, 0) sequence. Pin it (for now vertically) at coordinates (y + 1, y - 1). Progress by increasing n: for a given row n, if two adjacent yarns show side by side 0's, then cross them at this point.
%C A290691 Properties: n is a prime iff there is no 0 in row n. n is a square iff there is an isolated 0 in row n column 1. If there is a crossing between yarn y1 and yarn y2 in row n, then n = y1 * y2.
%C A290691 Alternative definition: row n is the list of (-n) mod (y) sorted in ascending order of abs(y - n / y), for all y candidate divisor of n, y between 2 and (n - 1) inclusive.
%F A290691 T(n,k) = (-n) mod y(n,k), with y(n,k) the yarn going through (n,k); ambiguity at a crossing doesn't matter, both mod yielding 0.
%e A290691 Array begins:
%e A290691 1
%e A290691 0 2
%e A290691 1 1 3
%e A290691 0 0 2 4
%e A290691 2 1 1 3 5
%e A290691 1 0 0 2 4 6
%e A290691 0 3 1 1 3 5 7
%e A290691 2 2 0 0 2 4 6 8
%e A290691 1 1 4 1 1 3 5 7 9
%e A290691 ...
%e A290691 Viewed as a braid (pairs of adjacent zeros being replaced by crossings):
%e A290691 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ------> k
%e A290691 .
%e A290691 3 1
%e A290691 |
%e A290691 4 0 2
%e A290691 | |
%e A290691 5 1 1 3
%e A290691 \ / |
%e A290691 6 0 2 4
%e A290691 / \ | |
%e A290691 7 2 1 1 3 5
%e A290691 | \ / | |
%e A290691 8 1 0 2 4 6
%e A290691 | / \ | | |
%e A290691 9 0 3 1 1 3 5 7
%e A290691 | | \ / | | |
%e A290691 10 2 2 0 2 4 6 8
%e A290691 | | / \ | | | |
%e A290691 11 1 1 4 1 1 3 5 7 9
%e A290691 \ / | \ / | | | |
%e A290691 12 0 3 0 2 4 6 8 10
%e A290691 / \ | / \ | | | | |
%e A290691 13 3 2 2 5 1 1 3 5 7 9 11
%e A290691 | | | | \ / | | | | |
%e A290691 14 2 1 1 4 0 2 4 6 8 10 12
%e A290691 | \ / | / \ | | | | | |
%e A290691 15 1 0 3 6 1 1 3 5 7 9 11 13
%e A290691 | / \ | | \ / | | | | | |
%e A290691 16 0 4 2 2 5 0 2 4 6 8 10 12 14
%e A290691 | | | | | / \ | | | | | | |
%e A290691 17 3 3 1 1 4 7 1 1 3 5 7 9 11 13 15
%e A290691 |
%e A290691 V ...
%e A290691 n
%t A290691 Ev[E_] := Module[{x, dx}, x = First[E]; dx = Last[E]; If[x == 0 && dx < 0, {-dx, -dx}, {x + dx, dx}]]
%t A290691 EvL[n_, L_] := Module[{LL}, LL = Ev /@ L; LL = Sort[LL]; LL = Append[LL, {n - 1, -1/n}]; LL]
%t A290691 It[nStart_, nEnd_, LStart_] := Module[{n, LL}, For[n = nStart; LL = LStart, n <= nEnd, n++, LL = EvL[n, LL]]; LL]
%t A290691 Encours[n_] := It[2, n, {}]
%t A290691 Countdown[x_, dx_] := If[dx > 0, (Ceiling[x] - x)/dx, (Floor[x] - x)/dx]
%t A290691 A[n_] := Drop[Apply[Countdown, #] & /@ Encours[n], -1]
%t A290691 Table[A[n], {n, 2, 25}] // Flatten
%t A290691 (* or *)
%t A290691 (Last /@ # &) /@ Sort /@ Table[{Abs[k - n/k], Mod[-n, k]}, {n, 3, 20}, {k, 2, n - 1}] // Flatten
%o A290691 (C)
%o A290691 #include <stdio.h>
%o A290691 #include <stdlib.h>
%o A290691 #include <string.h>
%o A290691 #define NMAX 40
%o A290691 struct cell { int f; int v; };
%o A290691 struct line { struct cell t[NMAX]; };
%o A290691 void display(struct line *T) { int n, k; for (n = 3; n <= NMAX; n ++) { for (k = 1; k < n - 1; k ++) { printf("%2d, ", T[n].t[k].v, T[n].t[k].f); } printf("\n"); } }
%o A290691 void swap(int *a, int *b) { int x; x = *a; *a = *b; *b = x; }
%o A290691 void fill(struct line *T)
%o A290691 {
%o A290691 int n, k;
%o A290691 for (n = 3; n <= NMAX; n ++)
%o A290691 {
%o A290691 for (k = 1; k < n - 2; k ++)
%o A290691 {
%o A290691 T[n].t[k].v = T[n - 1].t[k].v - 1;
%o A290691 T[n].t[k].f = T[n - 1].t[k].f;
%o A290691 }
%o A290691 T[n].t[n - 2].v = n - 2;
%o A290691 T[n].t[n - 2].f = n - 1;
%o A290691 for (k = 1; k < n - 2; k ++)
%o A290691 {
%o A290691 if ((T[n].t[k].v == 0) && (T[n].t[k + 1].v == 0))
%o A290691 {
%o A290691 swap(&T[n].t[k].f, &T[n].t[k + 1].f);
%o A290691 }
%o A290691 }
%o A290691 for (k = 1; k < n - 2; k ++)
%o A290691 {
%o A290691 if (T[n].t[k].v == -1)
%o A290691 {
%o A290691 T[n].t[k].v += T[n].t[k].f;
%o A290691 }
%o A290691 }
%o A290691 }
%o A290691 }
%o A290691 int main() { struct line T[NMAX + 1]; memset(T, 0x0, sizeof(T)); fill(T); display(T); }
%Y A290691 Cf. A293578.
%K A290691 nonn,tabl
%O A290691 1,3
%A A290691 _Luc Rousseau_, Oct 20 2017