A186328 - OEIS

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

%S 1,3,5,7,9,11,13,15,16,18,20,22,24,26,28,29,31,33,35,37,39,41,43,44,

%T 46,48,50,52,54,56,57,59,61,63,65,67,69,71,72,74,76,78,80,82,84,85,87,

%U 89,91,93,95,97,99,100,102,104,106,108,110,112,113,115,117,119,121,123,125,126,128,130,132,134,136,138,140,141,143,145,147,149,151,153,154,156,158,160,162,164,166,168,169,171,173,175,177,179,181,182,184,186

%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 pentagonal numbers and the hexagonal numbers.  Complement of A186329.

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

%e First, write

%e 1..5...12....22.....35......  (pentagonal)

%e 1....6....15....28.......45.. (hexagonal)

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

%e a=(1,3,5,7,9,11,13,15,16,....)=A186328

%e b=(2,4,6,8,10,12,14,17,19,...)=A186329.

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

%t d=1/2; u=3/2; v=-1/2; w=0; x=2; y=-1; 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}]  (* A186328 *)

%t Table[b[n], {n, 1, 100}]  (* A186329 *)

%Y Cf. A186219, A186329, A186330, A186331,

%Y A000384 (pentagonal), A000384 (hexagonal).

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 17 2011