A186541 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186542.

2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 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, 112, 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

Offset

1,1

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Links

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

Formula

a(n)=n+floor(sqrt((1/3)n^2+5/6))=A186541(n).

b(n)=n+floor(sqrt(3n^2-5/2))=A186542(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 after -2+3j^2:
a=(2,3,4,6,8,9,11,12,14,15,17,18,..)=A186541
b=(1,5,7,10,13,16,19,21,24,27,29...)=A186542.

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, A186539, A186540, A186542.

Sequence in context: A335878 A233133 A364216 * A207432 A020900 A333309

Adjacent sequences: A186538 A186539 A186540 * A186542 A186543 A186544

Keyword

nonn

Author

Clark Kimberling, Feb 23 2011

Status

approved