A186328 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 pentagonal numbers and the hexagonal numbers. Complement of A186329.

1, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 57, 59, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 125, 126, 128, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 164, 166, 168, 169, 171, 173, 175, 177, 179, 181, 182, 184, 186

Offset

1,2

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Links

Table of n, a(n) for n=1..100.

Example

First, write
1..5...12....22.....35......  (pentagonal)
1....6....15....28.......45.. (hexagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the hexagonal:
a=(1,3,5,7,9,11,13,15,16,....)=A186328
b=(2,4,6,8,10,12,14,17,19,...)=A186329.

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=3/2; v=-1/2; w=0; x=2; y=-1; 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}]  (* A186328 *)
Table[b[n], {n, 1, 100}]  (* A186329 *)

Crossrefs

Cf. A186219, A186329, A186330, A186331,

A000384 (pentagonal), A000384 (hexagonal).

Sequence in context: A120890 A321499 A134322 * A063460 A329831 A329938

Adjacent sequences: A186325 A186326 A186327 * A186329 A186330 A186331

Keyword

nonn

Author

Clark Kimberling, Feb 17 2011

Status

approved