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A186389
Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number). Complement of A186390.
4
4, 6, 8, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92
OFFSET
1,1
COMMENTS
See A186350 for a discussion of adjusted joint rank sequences.
EXAMPLE
First, write
......6.....12..18....24..30. (6i)
1..3..6..10...15....21..28... (triangular)
Then replace each number by its rank, where ties are settled by ranking 6i after the triangular:
a=(4,6,8,10,12,14,15,17,...)=A186389
b=(1,2,3,5,7,9,11,13,16,...)=A186390.
MATHEMATICA
(* adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
d=-1/2; u=6; v=0; x=1/2; y=1/2; (* 6i and triangular *)
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *)
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *)
Table[a[n], {n, 1, 120}] (* A186389 *)
Table[b[n], {n, 1, 100}] (* A186390 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 19 2011
STATUS
approved