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 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). 0
 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, 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, 1, 1, 4, 0, 0, 2, 4, 6, 8, 10, 12, 1, 0, 0, 3, 6, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. 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. 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. LINKS FORMULA 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. EXAMPLE Array begins: 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  6  8 1  1  4  1  1  3  5  7  9 ... Viewed as a braid (pairs of adjacent zeros being replaced by crossings):         1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  ------> k       .    3    1         |    4    0   2         |   |    5    1   1   3          \ /    |    6      0     2   4          / \    |   |    7    2   1   1   3   5         |    \ /    |   |    8    1     0     2   4   6         |    / \    |   |   |    9    0   3   1   1   3   5   7         |   |    \ /    |   |   |   10    2   2     0     2   4   6   8         |   |    / \    |   |   |   |   11    1   1   4   1   1   3   5   7   9          \ /    |    \ /    |   |   |   |   12      0     3     0     2   4   6   8  10          / \    |    / \    |   |   |   |   |   13    3   2   2   5   1   1   3   5   7   9  11         |   |   |   |    \ /    |   |   |   |   |   14    2   1   1   4     0     2   4   6   8  10  12         |    \ /    |    / \    |   |   |   |   |   |   15    1     0     3   6   1   1   3   5   7   9  11  13         |    / \    |   |    \ /    |   |   |   |   |   |   16    0   4   2   2   5     0     2   4   6   8  10  12  14         |   |   |   |   |    / \    |   |   |   |   |   |   |   17    3   3   1   1   4   7   1   1   3   5   7   9  11  13  15    |    V   ...    n MATHEMATICA Ev[E_] := Module[{x, dx}, x = First[E]; dx = Last[E]; If[x == 0 && dx < 0, {-dx, -dx}, {x + dx, dx}]] EvL[n_, L_] := Module[{LL}, LL = Ev /@ L; LL = Sort[LL]; LL = Append[LL, {n - 1, -1/n}]; LL] It[nStart_, nEnd_, LStart_] := Module[{n, LL}, For[n = nStart; LL = LStart, n <= nEnd, n++, LL = EvL[n, LL]]; LL] Encours[n_] := It[2, n, {}] Countdown[x_, dx_] := If[dx > 0, (Ceiling[x] - x)/dx, (Floor[x] - x)/dx] A[n_] := Drop[Apply[Countdown, #] & /@ Encours[n], -1] Table[A[n], {n, 2, 25}] // Flatten (* or *) (Last /@ # &) /@ Sort /@ Table[{Abs[k - n/k], Mod[-n, k]}, {n, 3, 20}, {k, 2, n - 1}] // Flatten PROG (C) #include #include #include #define NMAX 40 struct cell { int f; int v; }; struct line { struct cell t[NMAX]; }; 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"); } } void swap(int *a, int *b) { int x; x = *a; *a = *b; *b = x; } void fill(struct line *T) {     int n, k;     for (n = 3; n <= NMAX; n ++)     {         for (k = 1; k < n - 2; k ++)         {             T[n].t[k].v = T[n - 1].t[k].v - 1;             T[n].t[k].f = T[n - 1].t[k].f;         }         T[n].t[n - 2].v = n - 2;         T[n].t[n - 2].f = n - 1;         for (k = 1; k < n - 2; k ++)         {             if ((T[n].t[k].v == 0) && (T[n].t[k + 1].v == 0))             {                 swap(&T[n].t[k].f, &T[n].t[k + 1].f);             }         }         for (k = 1; k < n - 2; k ++)         {             if (T[n].t[k].v == -1)             {                 T[n].t[k].v += T[n].t[k].f;             }         }     } } int main() { struct line T[NMAX + 1]; memset(T, 0x0, sizeof(T)); fill(T); display(T); } CROSSREFS Cf. A293578. Sequence in context: A138948 A186114 A326934 * A155726 A325687 A230079 Adjacent sequences:  A290688 A290689 A290690 * A290692 A290693 A290694 KEYWORD nonn,tabl AUTHOR Luc Rousseau, Oct 20 2017 STATUS approved

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Last modified September 22 14:45 EDT 2020. Contains 337291 sequences. (Running on oeis4.)