A186342 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 octagonal numbers. Complement of A186343.

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 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, 88, 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, 158, 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..5...12....22..35..... (pentagonal)
1....8....21........40.. (octagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the octagonal:
a=(1,3,5,7,8,10,12,13,15,...)=A186342
b=(2,4,6,9,11,14,16,19,21,...)=A186343.

Mathematica

 (* adjusted joint ranking; general formula *)
d=1/2; u=3/2; 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}]  (* A186342 *)
Table[b[n], {n, 1, 100}]  (* A186343 *)

Crossrefs

Cf. A186219, A186343, A186344, A186345,

A000326 (pentagonal), A000567 (octagonal).

Sequence in context: A221056 A288467 A276224 * A186315 A285074 A186219

Adjacent sequences: A186339 A186340 A186341 * A186343 A186344 A186345

Keyword

nonn

Author

Clark Kimberling, Feb 18 2011

Status

approved