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A186354 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=3i and g(j)=j(j+1)/2 (triangular number).  Complement of A186355. 2
2, 4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See A186350.

LINKS

Table of n, a(n) for n=1..110.

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 before the triangular:

a=(2,4,6,8,9,11,12,14,15,17,....)=A186354

b=(1,3,5,7,10,13,16,20,24,28,...)=A186355.

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; (* odds 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}]  (* A186354 *)

Table[b[n], {n, 1, 100}]  (* A186355 *)

CROSSREFS

Cf. A186550, A186555, A186556, A186557.

Sequence in context: A191982 A080037 A184012 * A186149 A298861 A114571

Adjacent sequences:  A186351 A186352 A186353 * A186355 A186356 A186357

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 18 2011

STATUS

approved

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Last modified May 19 12:20 EDT 2022. Contains 353833 sequences. (Running on oeis4.)