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A186222 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares. Complement of A186221.

%I A186222 #11 Oct 18 2024 21:42:41
%S A186222 1,4,6,9,11,13,16,18,21,23,26,28,30,33,35,38,40,42,45,47,50,52,55,57,
%T A186222 59,62,64,67,69,71,74,76,79,81,83,86,88,91,93,96,98,100,103,105,108,
%U A186222 110,112,115,117,120,122,125,127,129,132,134,137,139,141,144,146,149,151,154,156,158,161,163,166,168,170,173,175,178,180,182,185,187,190,192,195,197,199,202,204,207,209,211,214,216,219,221,224,226,228,231,233,236,238,240
%N A186222 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and squares.  Complement of A186221.
%C A186222 See A186221.
%H A186222 G. C. Greubel, <a href="/A186222/b186222.txt">Table of n, a(n) for n = 1..10000</a>
%F A186222 a(n) = n + floor(-1/2 + sqrt(2*n^2)).
%e A186222 See A186221.
%t A186222 (* adjusted joint ranking; general formula *)
%t A186222 d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; z=0;
%t A186222 h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
%t A186222 a[n_]:=n+Floor[h[n]/(2x)];
%t A186222 k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
%t A186222 b[n_]:=n+Floor[k[n]/(2u)];
%t A186222 Table[a[n],{n,1,100}] (* A186221 *)
%t A186222 Table[b[n],{n,1,100}] (* A186222 *)
%t A186222 Table[n + Floor[Sqrt[2*n^2] - 1/2], {n, 1, 120}] (* _G. C. Greubel_, Aug 18 2018 *)
%o A186222 (PARI) vector(120, n, n + floor(-1/2 + sqrt(2*n^2))) \\ _G. C. Greubel_, Aug 18 2018
%o A186222 (Magma) [n + Floor(-1/2 + Sqrt(2*n^2)): n in [1..120]]; // _G. C. Greubel_, Aug 18 2018
%Y A186222 Cf. A186219, A186220, A186221.
%K A186222 nonn
%O A186222 1,2
%A A186222 _Clark Kimberling_, Feb 15 2011