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A186288 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 squares and pentagonal numbers. Complement of A186289.

%I A186288 #10 Feb 12 2025 03:14:44
%S A186288 1,3,5,7,9,11,12,14,16,18,20,21,23,25,27,29,31,32,34,36,38,40,41,43,
%T A186288 45,47,49,51,52,54,56,58,60,61,63,65,67,69,71,72,74,76,78,80,81,83,85,
%U A186288 87,89,90,92,94,96,98,100,101,103,105,107,109,110,112,114,116,118,120,121,123,125,127,129,130,132,134,136,138,140,141,143,145,147,149,150,152,154,156,158,160,161,163,165,167,169,170,172,174,176,178,179,181
%N A186288 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 squares and pentagonal numbers. Complement of A186289.
%C A186288 See A186219 for a discussion of adjusted rank sequences.
%e A186288 First, write
%e A186288 1..4...9....16....25..36..49..... (squares)
%e A186288 1....5...12....22....35......51.. (pentagonal)
%e A186288 Replace each number by its rank, where ties are settled by ranking the square number before the pentagonal:
%e A186288 a=(1,3,5,7,9,11,12,14,....)=A186288.
%e A186288 b=(2,4,6,8,10,13,15,17,...)=A186289.
%t A186288 (* adjusted joint ranking; general formula *)
%t A186288 d=1/2; u=1; v=0; w=0; x=3/2; y=-1/2; z=0;
%t A186288 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186288 a[n_]:=n+Floor[h[n]/(2x)];
%t A186288 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186288 b[n_]:=n+Floor[k[n]/(2u)];
%t A186288 Table[a[n],{n,1,100}]  (* A186288 *)
%t A186288 Table[b[n],{n,1,100}]  (* A186289 *)
%Y A186288 Cf. A186219, A186289, A186290, A186291, A000290 (squares), A000326 (pentagonal).
%K A186288 nonn
%O A186288 1,2
%A A186288 _Clark Kimberling_, Feb 17 2011