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A186223 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 pentagonal numbers. Complement of A186224.

%I A186223 #7 Sep 28 2019 22:25:02
%S A186223 1,3,5,6,8,9,11,13,14,16,17,19,20,22,24,25,27,28,30,31,33,35,36,38,39,
%T A186223 41,43,44,46,47,49,50,52,54,55,57,58,60,61,63,65,66,68,69,71,73,74,76,
%U A186223 77,79,80,82,84,85,87,88,90,91,93,95,96,98,99,101,102,104,106,107,109,110,112,114,115,117,118,120,121,123,125,126,128,129,131,132,134,136,137,139,140,142,143,145,147,148,150,151,153,155,156,158
%N A186223 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 pentagonal numbers.  Complement of A186224.
%C A186223 See A186219 for a general description.
%e A186223 First, write
%e A186223 1..3...6..10....15...21.....28......36...45...  (triangular)
%e A186223 1....5.........12...........22......35........... (pentagonal)
%e A186223 Replace each number by its rank, where ties are settled by ranking the triangular number before the pentagonal:
%e A186223 a=(1,3,5,6,8,9,11,13,...)
%e A186223 b=(2,4,7,10,12,15,18,...).
%t A186223 d=1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0;
%t A186223 (* triangular & pentagonal *)
%t A186223 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186223 a[n_]:=n+Floor[h[n]/(2x)];
%t A186223 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186223 b[n_]:=n+Floor[k[n]/(2u)];
%t A186223 Table[a[n],{n,1,100}]  (* A186223 *)
%t A186223 Table[b[n],{n,1,100}]  (* A186224 *)
%Y A186223 Cf. A186219, A186224, A186225, A186226,
%Y A186223 A000217 (triangular), A000326 (pentagonal).
%K A186223 nonn
%O A186223 1,2
%A A186223 _Clark Kimberling_, Feb 15 2011