A186227 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 heptagonal numbers. Complement of A186228.

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 109, 110, 111, 113, 114, 116, 117, 119, 120, 122, 123, 124, 126, 127, 129, 130, 132, 133, 135, 136, 138, 139, 140, 142, 143, 145

Offset

1,2

Comments

See A186219 for a general discussion of adjusted joint rank sequences.

Links

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

Example

First, write
1..3..6..10..15..21..28..36..45... (triangular)
1.......7......18......34.......55... (heptagonal)
Then replace each number by its rank, where ties are settled by ranking the triangular number before the heptagonal:
a=(1,3,4,6,7,9,10,12,...), A186227.
b=(2,5,8,11,15,18,21,...), A186228.

Mathematica

(* adjusted joint ranking; general formula *)
d=1/2; u=1/2; v=1/2; w=0; x=5/2; y=-3/2; 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}]  (* A186227 *)
Table[b[n], {n, 1, 100}]  (* A186228 *)

Crossrefs

Cf. A186219, A186228, A186237, A186238,

Cf. A000217 (triangular numbers)

Cf. A000566 (heptagonal numbers)

Sequence in context: A246443 A186495 A184746 * A258048 A185543 A026322

Adjacent sequences: A186224 A186225 A186226 * A186228 A186229 A186230

Keyword

nonn

Author

Clark Kimberling, Feb 16 2011

Status

approved