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.

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

Author

Luc Rousseau, Jun 20 2017

Keywords

• nonn
• tabf

Comments

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

Examples

			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
		

Links

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

Crossrefs

Cf. A053186, A288966, A293497, A293578.

Programs

Java

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);
}
}

Mathematica

(* 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 *)

PARI

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

Formula

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)

Extensions

More terms from Michel Marcus, Jun 21 2017