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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
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
LINKS
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)
EXAMPLE
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
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 *)
PROG
(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);
}
}
(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
CROSSREFS
Sequence in context: A262709 A107628 A268389 * A305355 A218380 A152815
KEYWORD
nonn,tabf
AUTHOR
Luc Rousseau, Jun 20 2017
EXTENSIONS
More terms from Michel Marcus, Jun 21 2017
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)