A186159 - OEIS

%I #7 Mar 30 2012 18:57:18

%S 1,3,4,6,7,8,10,11,13,14,16,17,18,20,21,23,24,25,27,28,30,31,32,34,35,

%T 37,38,39,41,42,44,45,47,48,49,51,52,54,55,56,58,59,61,62,63,65,66,68,

%U 69,70,72,73,75,76,77,79,80,82,83,85,86,87,89,90,92,93,94,96,97,99,100,101,103,104,106,107,108,110,111,113,114,116,117,118,120,121,123,124,125,127,128,130,131,132,134,135,137,138,139,141

%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 triangular numbers and octagonal numbers.  Complement of A186274.

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

%e First, write the triangular and octagonal numbers:

%e 1..3..6.....10..15..21..28

%e 1........8..........21......

%e Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal:

%e a=(1,3,4,6,7,8,10,11,13,...)=A186159.

%e b=(2,5,9,12,15,19,22,26,...)=A186274.

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

%t d=1/2; u=1/2; v=1/2; 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}] (* A186159 *)

%t Table[b[n],{n,1,100}] (* A186274 *)

%Y Cf. A186219, A186274, A186275, A186276,

%Y Cf. A000217 (triangular numbers).

%Y Cf. A000567 (octagonal numbers).

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 13 2011