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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.
6
1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 26, 28, 30, 33, 35, 38, 40, 42, 45, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 71, 74, 76, 79, 81, 83, 86, 88, 91, 93, 96, 98, 100, 103, 105, 108, 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
OFFSET
1,2
COMMENTS
See A186221.
LINKS
FORMULA
a(n) = n + floor(-1/2 + sqrt(2*n^2)).
EXAMPLE
See A186221.
MATHEMATICA
(* adjusted joint ranking; general formula *)
d=-1/4; u=1/2; v=1/2; w=0; x=1; y=0; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}] (* A186221 *)
Table[b[n], {n, 1, 100}] (* A186222 *)
Table[n + Floor[Sqrt[2*n^2] - 1/2], {n, 1, 120}] (* G. C. Greubel, Aug 18 2018 *)
PROG
(PARI) vector(120, n, n + floor(-1/2 + sqrt(2*n^2))) \\ G. C. Greubel, Aug 18 2018
(Magma) [n + Floor(-1/2 + Sqrt(2*n^2)): n in [1..120]]; // G. C. Greubel, Aug 18 2018
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Clark Kimberling, Feb 15 2011
STATUS
approved