A186223 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 pentagonal numbers. Complement of A186224.

1, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158

Offset

1,2

Comments

See A186219 for a general description.

Links

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

Example

First, write
1..3...6..10....15...21.....28......36...45...  (triangular)
1....5.........12...........22......35........... (pentagonal)
Replace each number by its rank, where ties are settled by ranking the triangular number before the pentagonal:
a=(1,3,5,6,8,9,11,13,...)
b=(2,4,7,10,12,15,18,...).

Mathematica

d=1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0;
(* triangular & pentagonal *)
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}]  (* A186223 *)
Table[b[n], {n, 1, 100}]  (* A186224 *)

Crossrefs

Cf. A186219, A186224, A186225, A186226,

A000217 (triangular), A000326 (pentagonal).

Sequence in context: A186324 A101358 A276210 * A047446 A058065 A218773

Adjacent sequences: A186220 A186221 A186222 * A186224 A186225 A186226

Keyword

nonn

Author

Clark Kimberling, Feb 15 2011

Status

approved