A186159 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and octagonal numbers. Complement of A186274.

1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 113, 114, 116, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 132, 134, 135, 137, 138, 139, 141

Offset: 1

Author

Clark Kimberling, Feb 13 2011

Keywords

• nonn

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write the triangular and octagonal numbers:
1..3..6.....10..15..21..28
1........8..........21......
Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal:
a=(1,3,4,6,7,8,10,11,13,...)=A186159.
b=(2,5,9,12,15,19,22,26,...)=A186274.
		

Crossrefs

Cf. A186219, A186274, A186275, A186276,

Cf. A000217 (triangular numbers).

Cf. A000567 (octagonal numbers).

Programs

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1/2; v=1/2; w=0; x=3; y=-2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n],{n,1,100}] (* A186159 *)
Table[b[n],{n,1,100}] (* A186274 *)