Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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