

A186539


Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=2+3j^2. Complement of A186540.


4



1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108, 110, 111, 113, 115, 116, 118, 119, 121, 123, 124, 126, 127, 129, 130, 132, 134, 135, 137, 138, 140, 141, 143, 145, 146, 148, 149, 151, 153, 154, 156, 157
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OFFSET

1,2


COMMENTS

See A186219 for a discussion of adjusted joint rank sequences.
Differs from A059555 at n=97, 123, 194, 220, 246, ...  R. J. Mathar, May 18 2020


LINKS

Table of n, a(n) for n=1..100.


FORMULA

a(n)=n+floor(sqrt((1/3)n^2+1/24))=A186539(n).
b(n)=n+floor(sqrt(3n^23/2))=A186540(n).


EXAMPLE

First, write
1..4..9..16..25..36..49.... (i^2)
.......10....25.....46.. (2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before 2+3j^2:
a=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539
b=(2,5,8,10,13,16,19,21,24,27,30...)=A186540.


MATHEMATICA

(* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)
d = 1/2; u = 1; v = 0; w = 0; x = 3; y = 0; z = 2;
h[n_] := y + (4 x (u*n^2 + v*n + w  z  d) + y^2)^(1/2);
a[n_] := n + Floor[h[n]/(2 x)];
k[n_] := v + (4 u (x*n^2 + y*n + z  w + d) + v^2)^(1/2);
b[n_] := n + Floor[k[n]/(2 u)];
Table[a[n], {n, 1, 100}] (* A186539 *)
Table[b[n], {n, 1, 100}] (* A186540 *)


CROSSREFS

Cf. A186219, A186540, A186541, A186542.
Sequence in context: A243989 A248521 A059555 * A054385 A284773 A172272
Adjacent sequences: A186536 A186537 A186538 * A186540 A186541 A186542


KEYWORD

nonn


AUTHOR

Clark Kimberling, Feb 23 2011


STATUS

approved



