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A186356 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number). Complement of A186357. 2
3, 5, 6, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 143, 144, 145, 146 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
EXAMPLE
First, write
...3..6..9....12..15..18..21..24.. (3*i)
1..3..6....10.....15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking 3i after the triangular:
a=(3,5,6,8,10,11,13,14,15,..)=A186356
b=(1,2,4,7,9,12,16,19,23,...)=A186357.
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=3; v=0; x=1/2; y=1/2;
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}] (* A186356 *)
Table[b[n], {n, 1, 100}] (* A186357 *)
CROSSREFS
Sequence in context: A354140 A183872 A288934 * A184582 A184734 A189662
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 18 2011
STATUS
approved

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Last modified March 3 04:18 EST 2024. Contains 370499 sequences. (Running on oeis4.)