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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 Luc Rousseau, The first 25 lines of the triangle array, formatted 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 Cf. A053186, A288966, A293497, A293578. Sequence in context: A262709 A107628 A268389 * A305355 A218380 A152815 Adjacent sequences:  A288966 A288967 A288968 * A288970 A288971 A288972 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 October 22 07:06 EDT 2021. Contains 348160 sequences. (Running on oeis4.)