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A186326 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and octagonal numbers. Complement of A186327.

%I A186326 #8 Apr 20 2018 00:39:16
%S A186326 2,3,5,6,8,9,11,12,14,16,17,19,20,22,24,25,27,28,30,31,33,35,36,38,39,
%T A186326 41,42,44,46,47,49,50,52,53,55,57,58,60,61,63,65,66,68,69,71,72,74,76,
%U A186326 77,79,80,82,83,85,87,88,90,91,93,94,96,98,99,101,102,104,106,107,109,110,112,113,115,117,118,120,121,123,124,126,128,129,131,132,134,135,137,139,140,142,143,145,147,148,150,151,153,154,156,158
%N A186326 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and octagonal numbers.  Complement of A186327.
%H A186326 Matthias Christandl, Fulvio Gesmundo, Asger Kjærulff Jensen, <a href="https://arxiv.org/abs/1801.04852">Border rank is not multiplicative under the tensor product</a>, arXiv:1801.04852 [math.AG], 2018.
%e A186326 First, write
%e A186326 1..4...9..16....25..36....49..64...  (squares)
%e A186326 1....8.......21........40........65. (octagonal)
%e A186326 Replace each number by its rank, where ties are settled by ranking the square number after the octagonal:
%e A186326 a=(2,3,5,6,8,9,11,12,14,...)=A186326
%e A186326 b=(1,4,7,10,13,15,18,21,...)=A186327.
%t A186326 (* adjusted joint ranking; general formula *)
%t A186326 d=-1/2; u=1; v=0; w=0; x=3; y=-2; z=0;
%t A186326 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186326 a[n_]:=n+Floor[h[n]/(2x)];
%t A186326 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186326 b[n_]:=n+Floor[k[n]/(2u)];
%t A186326 Table[a[n], {n, 1, 100}]  (* A186326 *)
%t A186326 Table[b[n], {n, 1, 100}]  (* A186327 *)
%Y A186326 Cf. A186219, A186324, A186325, A186327.
%K A186326 nonn
%O A186326 1,1
%A A186326 _Clark Kimberling_, Feb 17 2011