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%I #5 Mar 30 2012 18:57:18
%S 1,3,5,7,8,10,12,13,15,17,18,20,22,24,25,27,29,30,32,34,36,37,39,41,
%T 42,44,46,48,49,51,53,54,56,58,59,61,63,65,66,68,70,71,73,75,77,78,80,
%U 82,83,85,87,88,90,92,94,95,97,99,100,102,104,106,107,109,111,112,114,116,118,119,121,123,124,126,128,129,131,133,135,136,138,140,141,143,145,147,148,150,152,153,155,157,158,160,162,164,165,167,169,170
%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 octagonal numbers. Complement of A186343.
%C See A186219 for a discussion of adjusted joint rank sequences.
%e First, write
%e 1..5...12....22..35..... (pentagonal)
%e 1....8....21........40.. (octagonal)
%e Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the octagonal:
%e a=(1,3,5,7,8,10,12,13,15,...)=A186342
%e b=(2,4,6,9,11,14,16,19,21,...)=A186343.
%t (* adjusted joint ranking; general formula *)
%t d=1/2; u=3/2; 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}] (* A186342 *)
%t Table[b[n], {n, 1, 100}] (* A186343 *)
%Y Cf. A186219, A186343, A186344, A186345,
%Y A000326 (pentagonal), A000567 (octagonal).
%K nonn
%O 1,2
%A _Clark Kimberling_, Feb 18 2011
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