A293578 -id:A293578 - cp's OEIS frontend

A288969 Triangular array read by rows: row n is the list of the 2n-1 successive values taken by the function z = n - floor(x) floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.

0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 0, 2, 1, 0, 0, 1, 2, 3, 1, 3, 2, 1, 0, 0, 1, 2, 3, 0, 2, 0, 3, 2, 1, 0, 0, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 0, 2, 4, 2, 0, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 1, 3, 0, 3, 1, 5, 4, 3, 2, 1, 0

Offset: 1

Luc Rousseau, Jun 20 2017

See A288966's links for explanations about the algorithm used to go along an hyperbola of equation y = n/x, with 1 <= x <= n.
When represented as a triangular array, internal zeros "0" correspond to factorizations of n.
This array appears to resemble a version of the sieve of Eratosthenes with zeros aligned.
A053186 and A293497 appear to intertwine into this sequence. The following will be denoted "assumption (1)": with t indexing columns, t=0 being central: T(n, 2k) = A053186(n+k^2) and T(n, 2k+1) = A293497(n+k(k+1)). - Luc Rousseau, Oct 11 2017
It would be nice to have a larger b-file, or an a-file. - N. J. A. Sloane, Oct 13 2017

			Array begins:
                0
              0 1 0
            0 1 2 1 0
          0 1 2 0 2 1 0
        0 1 2 3 1 3 2 1 0
      0 1 2 3 0 2 0 3 2 1 0
    0 1 2 3 4 1 3 1 4 3 2 1 0
  0 1 2 3 4 0 2 4 2 0 4 3 2 1 0

Luc Rousseau, The first 25 lines of the triangle array, formatted

Cf. A053186, A288966, A293497, A293578.

package oeis;
public class B {
public static void main(String[] args) {
for (int n = 1; n <= 8; n ++) {
hyberbolaTiles(n);
}
}
private static void hyberbolaTiles(int n) {
int x = 0, y = 0, p = 0, q = n;
do {
if (p != 0) {
System.out.println(n - p * q);
}
if (y < 0) { x = y + q; q --; }
if (y > 0) { p ++; x = y - p; }
if (y == 0) {
p ++;
x = 0; System.out.println("0");
q --;
}
y = x + p - q;
} while (q > 0);
}
}

(* Under assumption (1) *)
A288969[n_, t_] := Module[{x},
  x = Floor[(-t + Sqrt[t^2 + 4 n])/2];
  n - x (t + x)
] (* Luc Rousseau, Oct 11 2017 *)
(* or *)
FEven[x_] := x^ 2
InvFEven[x_] := Sqrt[x]
GEven[n_] := n - FEven[Floor[InvFEven[n]]]
FOdd[x_] := x*(x + 1)
InvFOdd[x_] := (Sqrt[1 + 4 x] - 1)/2
GOdd[n_] := n - FOdd[Floor[InvFOdd[n]]]
A288969[n_, t_] := Module[
  {e, k, x},
  e = EvenQ[t];
  k = If[e, t/2, (t - 1)/2];
  x = n + If[e, FEven[k], FOdd[k]];
  If[e, GEven[x], GOdd[x]]
] (* Luc Rousseau, Oct 11 2017 *)

htrow(n) = {my(x = 0, y = 0, p = 0, q = n); while (q>0, if (p, print1(n-p*q, ", ")); if (y < 0, x = y + q; q --); if (y > 0, p ++; x = y - p); if (y == 0, p++; x = 0; print1(0, ", "); q --;); y = x + p - q;);}
tabf(nn) = for (n=1, nn, htrow(n); print()); \\ Michel Marcus, Jun 21 2017

From Luc Rousseau, Oct 11 2017: (Start)
(All formulas under assumption (1))
With t indexing columns, t=0 being central,
T(n, 2k) = A053186(n+k^2).
T(n, 2k+1) = A293497(n+k(k+1)).
T(n, t) = n - x*(x+t) where x = floor((-t+sqrt(t^2+4n))/2).
With A293578 viewed as a 2D array T',
T'(n,t)=T(n-1,t)-T(n,t)+1 (define T(0,0) as 0).
(End)

More terms from Michel Marcus, Jun 21 2017

A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).

Original entry on oeis.org

1, -1, 0, 2, 1, 0, 2, -1, 1, 2, 0, 3, 1, 1, 2, 0, 3, -1, 2, 2, 1, 3, 0, 4, 1, 2, 2, 1, 3, 0, 4, -1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 3, 2, 2, 3, 1, 4, 0, 5, -1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, -1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, -1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, -1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, -1

Offset: 1

Luc Rousseau, Oct 14 2017

Numeric characterization:
Row n is the value of a list after n iterations of the following algorithm:
- start with an empty list (assimilable to row number 0)
- Iteration n consists of
-- if n is odd, appending 1 to the left of the list and -1 to the right;
-- if n is even, replacing each value in the list by its complement to n/2.
Underlying definition and interest: this sequence represents a square array in which each cell is a structure made of 4 values arranged in W/N/S/E fashion. These values are twice the areas of elementary right triangles that enter the composition of quadrilaterals delimited by two families of lines, with the following equations:
- for m = 1, 2, 3, ...: y = mx - (m-1)^2 {x <= m-1}
- for n = -1, 0, 1, ...: y = -nx - (n+1)^2 {x >= 1-n}
Globally these quadrilaterals form an anamorphosis of A002024. See provided link for explanations and illustrations.

			Array begins (characterization)(x stands for -1):
              1 x
              0 2
            1 0 2 x
            1 2 0 3
          1 1 2 0 3 x
          2 2 1 3 0 4
        1 2 2 1 3 0 4 x
        3 2 2 3 1 4 0 5
      1 3 2 2 3 1 4 0 5 x
      4 2 3 3 2 4 1 5 0 6
    1 4 2 3 3 2 4 1 5 0 6 x
    5 2 4 3 3 4 2 5 1 6 0 7
  1 5 2 4 3 3 4 2 5 1 6 0 7 x
Or (definition)(to be read by antidiagonals):
    x     x     x     x
  1   2 2   3 3   4 4   5 ...
    0     0     0     0
    0     0     0     0
  1   2 2   3 3   4 4   5 ...
    1     1     1     1
    1     1     1     1
  1   2 2   3 3   4 4   5 ...
    2     2     2     2
    2     2     2     2
  1   2 2   3 3   4 4   5 ...
    3     3     3     3
    3     3     3     3
  1   2 2   3 3   4 4   5 ...
    4     4     4     4
  ...

Luc Rousseau, Relation between A293670 and A002024 - Numeric structure of an anamorphosis

Cf. A293578, A002024.

evolve(L,n)=if(n%2==1,listinsert(L,1,1);listinsert(L,-1,#L+1),L=apply(v->n/2-v,L));L
N=30;L=List();for(n=1,N,L=evolve(L,n);for(i=1,#L,print1(L[i],", "));print())

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).

Original entry on oeis.org

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

Offset: 1

Luc Rousseau, Oct 20 2017

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.

			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

Cf. A293578.

#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); }

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

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.

cp's OEIS Frontend

A288969 Triangular array read by rows: row n is the list of the 2n-1 successive values taken by the function z = n - floor(x) floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.

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A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).

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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).

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cp's OEIS Frontend

A288969 Triangular array read by rows: row n is the list of the 2*n-1 successive values taken by the function z = n - floor(x) * floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.

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Java

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A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

C

Mathematica

Formula

A288969 Triangular array read by rows: row n is the list of the 2n-1 successive values taken by the function z = n - floor(x) floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.