A186324 - OEIS

%I #6 Mar 30 2012 18:57:18

%S 1,3,5,6,8,9,11,12,14,16,17,19,20,22,23,25,27,28,30,31,33,35,36,38,39,

%T 41,42,44,46,47,49,50,52,53,55,57,58,60,61,63,65,66,68,69,71,72,74,76,

%U 77,79,80,82,83,85,87,88,90,91,93,94,96,98,99,101,102,104,106,107,109,110,112,113,115,117,118,120,121,123,124,126,128,129,131,132,134,135,137,139,140,142,143,145,147,148,150,151,153,154,156,158

%N 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 octagonal numbers.  Complement of A186325.

%C See A186219 for a discussion of adjusted joint rank sequences.

%e First, write

%e 1..4...9..16....25..36....49..64...  (squares)

%e 1....8.......21........40........65. (octagonal)

%e Replace each number by its rank, where ties are settled by ranking the square number before the octagonal:

%e a=(1,3,5,6,8,9,11,12,14,...)=A186324

%e b=(2,4,7,10,13,15,18,21,...)=A186325.

%t (* adjusted joint ranking; general formula *)

%t d=1/2; u=1; v=0; w=0; x=3; y=-2; z=0;

%t h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);

%t a[n_]:=n+Floor[h[n]/(2x)];

%t k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);

%t b[n_]:=n+Floor[k[n]/(2u)];

%t Table[a[n], {n, 1, 100}]  (* A186324 *)

%t Table[b[n], {n, 1, 100}]  (* A186325 *)

%Y Cf. A186219, A186325, A186326, A186327,

%Y A000290 (squares), A000567 (octagonal).

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 17 2011