A181118 Sequencing of all rational numbers p/q > 0 as ordered pairs (p,q). The rational (p,q) occurs as the n-th ordered pair where n=(p+q-1)*(p+q-2)/2+q.

1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 1, 3, 4, 1, 3, 2, 2, 3, 1, 4, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3

Offset: 1

Author

Frank M Jackson, Oct 04 2010

Keywords

• easy
• nonn
• tabf

Comments

From L. Edson Jeffery, Dec 17 2011: (Start)

Arrange the ordered pairs in rows

(1,1)

(2,1),(1,2)

(3,1),(2,2),(1,3)

etc., and let the rows be indexed by n=1,2,.... Then the sum of the products of the pairs in row n is equal to A000292(n). For example, for n=3, 3*1+2*2+1*3=A000292(3)=10. (End)

Examples

			Triangle begins:
1,1                  : 1/1;
2,1,1,2              : 2/1, 1/2;
3,1,2,2,1,3          : 3/1, 2/2, 1/3;
4,1,3,2,2,3,1,4      : 4/1, 3/2, 2/3, 1/4;
5,1,4,2,3,3,2,4,1,5  : 5/1, 4/2, 3/3, 2/4, 1/5;
...
		

Links

• G. H. Hardy, A Course of Pure Mathematics (1921), p. 1.

Crossrefs

Cf. A000292, A057555.

Programs

Mathematica

Flatten[Table [{n+1-r, r}, {n, 9}, {r, n}]]
u[x_] := Floor[3/2 + Sqrt[2*x]]; v[x_] := Floor[1/2 + Sqrt[2*x]]; n[x_] := 1 - x + u[x]*(u[x] - 1)/2; k[x_] := x - v[x]*(v[x] - 1)/2; Flatten[Table[{n[m], k[m]}, {m, 45}]] (* L. Edson Jeffery, Jun 20 2015 *)

PARI

for(n=1,9,for(r=1,n,print1(n+1-r", "r", "))) \\ Charles R Greathouse IV, Dec 20 2011

Formula

Triangle format R(n,m) of ordered pairs (R(n,2r-1), R(n,2r)) with R(n,2r-1)=n+1-r and R(n,2r)=r and generating the rational (n+1-r)/r.

Extensions

Typo corrected and tabl changed to tabf by Frank M Jackson, Oct 07 2010