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A186384
Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=5i and g(j)=j(j+1)/2 (triangular number). Complement of A186383.
6
1, 2, 4, 6, 8, 10, 12, 15, 18, 21, 24, 27, 31, 35, 39, 43, 47, 52, 57, 62, 67, 72, 78, 84, 90, 96, 102, 109, 116, 123, 130, 137, 145, 153, 161, 169, 177, 186, 195, 204, 213, 222, 232, 242, 252, 262, 272, 283, 294, 305, 316, 327, 339, 351, 363, 375, 387, 400, 413, 426, 439, 452, 466, 480, 494, 508, 522, 537, 552, 567, 582, 597, 613, 629, 645, 661, 677, 694, 711, 728, 745, 762, 780, 798, 816, 834, 852, 871, 890
OFFSET
1,2
EXAMPLE
First, write
.....5...10..15..20..25..30.. (5i)
1..3..6..10..15....21..28.. (triangular)
Then replace each number by its rank, where ties are settled by ranking 5i before the triangular:
a=(3,5,7,9,11,13,14,16,17,..)=A186383
b=(1,2,4,6,8,10,12,15,18,...)=A186384.
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=5; v=0; x=1/2; y=1/2; (* 5i 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}] (* A186383 *)
Table[b[n], {n, 1, 100}] (* A186384 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 19 2011
STATUS
approved