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A186219 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 squares. Complement of A186220.

%I A186219 #13 Sep 08 2022 08:45:55
%S A186219 1,3,5,7,8,10,12,13,15,17,19,20,22,24,25,27,29,31,32,34,36,37,39,41,
%T A186219 43,44,46,48,49,51,53,54,56,58,60,61,63,65,66,68,70,72,73,75,77,78,80,
%U A186219 82,83,85,87,89,90,92,94,95,97,99,101,102,104,106,107,109,111,113,114,116,118,119,121,123,124,126,128,130,131,133,135,136,138,140,142,143,145,147,148,150,152,153,155,157,159,160,162,164,165,167,169,171
%N A186219 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 squares.  Complement of A186220.
%C A186219 Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences.  If 0<|d|<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint.  Let f^=(inverse of f) and g^=(inverse of g).  When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))-d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d).  Therefore, the sequences a and b are a complementary pair.
%C A186219 Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
%C A186219 (1) if 0<d<1, we call a and b the "adjusted joint rank sequences of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j)"; (2) if -1<d<0, we call a and b the "adjusted joint rank sequences of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j)".
%C A186219 Using f(i)=ui^2+vi+w and g(j)=xj^2+yj+z, we can carry out adjusted joint rankings of any pair of polygonal sequences (triangular, square, pentagonal, etc.)  In this case,
%C A186219   a(n)=n+floor((-y+sqrt(4x(un^2+vn+w-z-d)+y^2))/(2x)),
%C A186219   b(n)=n+floor((-v+sqrt(4u(xn^2+yn+z-w+d)+v^2)/(2u)),
%C A186219   where a(n) is the rank of un^2+vn+w and b(n) is the rank
%C A186219   of xn^2+yn+z+d, where d must be chosen small enough, in
%C A186219   absolute value, that the sets F and G are disjoint.
%C A186219 Example:  f=A000217 (triangular numbers) and g=A000290 (squares) yield adjusted rank sequences a=A186219 and b=A186220 for d=1/4 and a=A186221 and b=A186222 for d=-1/4.
%H A186219 G. C. Greubel, <a href="/A186219/b186219.txt">Table of n, a(n) for n = 1..10000</a>
%F A186219 a(n) = n + floor(sqrt((n^2+n)/2 - 1/4)), (A186219).
%F A186219 b(n) = n + floor((-1 + sqrt(8*n^2+3))/2), (A186220).
%e A186219 First, write
%e A186219 1..3...6..10..15...21..28..36..45...  (triangular)
%e A186219 1....4.. 9......16...25....36....49.. (square)
%e A186219 Replace each number by its rank, where ties are settled by ranking the triangular number before the square:
%e A186219 a=(1,3,5,7,8,10,12,13,...)
%e A186219 b=(2,4,6,9,11,14,16,18,...).
%t A186219 (* adjusted joint ranking of triangular numbers and squares, using general formula *)
%t A186219 d=1/4; u=1/2; v=1/2; w=0; x=1; y=0; z=0;
%t A186219 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186219 a[n_]:=n+Floor[h[n]/(2x)]; (* rank of triangular n(n+1)/2 *)
%t A186219 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186219 b[n_]:=n+Floor[k[n]/(2u)]; (* rank of square n^2 *)
%t A186219 Table[a[n],{n,1,100}] (* A186219 *)
%t A186219 Table[b[n],{n,1,100}] (* A186220 *)
%o A186219 (PARI) vector(100, n, n + floor(sqrt((n^2 + n)/2 - 1/4))) \\ _G. C. Greubel_, Aug 26 2018
%o A186219 (Magma) [n + Floor(Sqrt((n^2 + n)/2 - 1/4)): n in [1..100]]; // _G. C. Greubel_, Aug 26 2018
%Y A186219 Cf. A186145 (joint ranks of squares and cubes),
%Y A186219 A000217 (triangular numbers),
%Y A186219 A000290 (squares),
%Y A186219 A186220 (complement of A186119)
%Y A186219 A186221 ("after" instead of "before")
%Y A186219 A186222 (complement of A186221).
%K A186219 nonn
%O A186219 1,2
%A A186219 _Clark Kimberling_, Feb 15 2011