

A186350


Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the odd numbers and the triangular numbers. Complement of A186351.


20



1, 3, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 139, 140, 141
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OFFSET

1,2


COMMENTS

Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences. If 0<d<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint. Let f^=(inverse of f) and g^=(inverse of g). When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d). Therefore, the sequences a and b are a complementary pair.
Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
(1) if 0<d<1, we call a and b the "adjusted joint rank sequences of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j)"; (2) if 1<d<0, we call a and b the "adjusted joint rank sequences of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j)".
Using f(i)=ui+v, g(j)=xj^2+yj+z, we find a and b given by
a(n)=n+floor((y+sqrt(4x(un+vd)+y^2))/(2x)),
b(n)=n+floor((xn^2+ynv+d)/(2u))),
where a(n) is the rank of un+v and b(n) is the rank
xn^2+yn+z+d, and d must be chosen small enough, in
absolute value, that the sets F and G are disjoint.
Example: f=A000217 (odd numbers) and g=A000290 (triangular numbers) yield adjusted joint rank sequences a=A186350 and b=A186351 for d=1/2 and a=A186352 and b=A186353 for d=1/2.
For other classes of adjusted joint rank sequences, see A186145 and A186219.


LINKS

Table of n, a(n) for n=1..120.


FORMULA

a(n)=n+floor(1/2+sqrt(4n9/4))=A186350(n).
b(n)=n+floor((n^2+n+3)/4)=A186351(n).


EXAMPLE

First, write
1..3..5..7..9..11..13..15..17..21..23.. (odds)
1..3....6.....10.......15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking the odd number before the triangjular:
a=(1,3,5,7,8,10,11,12,14,....)=A186350
b=(2,4,6,9,13,17,21,26,32,...)=A186351.


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=2; v=1; x=1/2; y=1/2; (* odds and triangular *)
h[n_]:=(y+(4x(u*n+vd)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *)
k[n_]:=(x*n^2+y*nv+d)/u;
b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *)
Table[a[n], {n, 1, 120}] (* A186350 *)
Table[b[n], {n, 1, 100}] (* A186351 *)


CROSSREFS

Cf. A186145, A186219, A186351, A186352, A186353,
A005408 (odd numbers), A000217 (triangular numbers).
Sequence in context: A212294 A299495 A186689 * A189829 A175146 A171508
Adjacent sequences: A186347 A186348 A186349 * A186351 A186352 A186353


KEYWORD

nonn


AUTHOR

Clark Kimberling, Feb 18 2011


STATUS

approved



