A186227 - OEIS

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

%S 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,

%T 38,39,41,42,43,45,46,48,49,51,52,54,55,56,58,59,61,62,64,65,67,68,69,

%U 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

%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 heptagonal numbers.  Complement of A186228.

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

%e First, write

%e 1..3..6..10..15..21..28..36..45... (triangular)

%e 1.......7......18......34.......55... (heptagonal)

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

%e a=(1,3,4,6,7,9,10,12,...), A186227.

%e b=(2,5,8,11,15,18,21,...), A186228.

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

%t d=1/2; u=1/2; v=1/2; w=0; x=5/2; y=-3/2; 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}]  (* A186227 *)

%t Table[b[n],{n,1,100}]  (* A186228 *)

%Y Cf. A186219, A186228, A186237, A186238,

%Y Cf. A000217 (triangular numbers)

%Y Cf. A000566 (heptagonal numbers)

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 16 2011