A186539 - OEIS

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.

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^2-3/2))=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 A375619

Adjacent sequences: A186536 A186537 A186538 * A186540 A186541 A186542

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 23 2011

STATUS

approved