A186315 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 hexagonal numbers. Complement of A186316.

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111, 112, 114, 116, 118, 119, 121, 123, 124, 126, 128, 129, 131, 133, 135, 136, 138, 140, 141, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 170

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..4...9...16..25....36....49. (squares)
1....6...15.......28....45.... (hexagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the hexagonal:
a=(1,3,5,7,8,10,12,13,...)=A186315.
b=(2,4,6,9,11,14,16,18,...)=A186316.

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; 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}]  (* A186315 *)
Table[b[n], {n, 1, 100}]  (* A186316 *)

Crossrefs

Cf. A186219, A186316, A186317, A186318,

A000290 (squares), A000384 (hexagonal numbers).

Sequence in context: A288467 A276224 A186342 * A285074 A186219 A185050

Adjacent sequences: A186312 A186313 A186314 * A186316 A186317 A186318

Keyword

nonn

Author

Clark Kimberling, Feb 17 2011

Status

approved