A186288 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 squares and pentagonal numbers. Complement of A186289.

1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 120, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 150, 152, 154, 156, 158, 160, 161, 163, 165, 167, 169, 170, 172, 174, 176, 178, 179, 181

Offset

1,2

Comments

See A186219 for a discussion of adjusted rank sequences.

Links

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

Example

First, write
1..4...9....16....25..36..49..... (squares
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the pentagonal:
a=(1,3,5,7,9,11,12,14,....)=A186288.
b=(2,4,6,8,10,13,15,17,...)=A186289.

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=3/2; y=-1/2; 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}]  (* A186288 *)
Table[b[n], {n, 1, 100}]  (* A186289 *)

Crossrefs

Cf. A186219, A186289, A186290, A186291,

A000290 (squares), A000326 (pentagonal).

Sequence in context: A186346 A248633 A137228 * A272915 A094391 A158919

Adjacent sequences: A186285 A186286 A186287 * A186289 A186290 A186291

Keyword

nonn

Author

Clark Kimberling, Feb 17 2011

Status

approved