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A186499 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)=-4+5j^2.  Complement of A186500. 6
1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 109, 111, 112, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 133, 134, 136, 137, 138, 140, 141, 143, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A186219 for a discussion of adjusted joint rank sequences.

The pairs (i,j) for which i^2=-4+5j^2 are (L(2h-2),F(2h-1)), where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers); compare this with the comment at A186511.

LINKS

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

FORMULA

a(n)=n+floor((1/10)(sqrt(2n^2+7)))=A186499(n).

b(n)=n+floor(sqrt(5n^2-7/2))=A186500(n).

EXAMPLE

First, write

1..4..9..16..25..36..49..... (i^2)

1........16........41........(-4+5j^2)

Then replace each number by its rank, where ties are settled by ranking i^2 before -4+5j^2:

a=(1,3,4,5,7,8,10,11,13,14,15,17,18...)=A186499

b=(2,6,9,12,16,19,22,25,29,32,35,38,..)=A186500.

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=5; y=0; z=4;

h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);

a[n_]:=n+Floor[h[n]/(2 x)];

k[n_]:=-v+(4u(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}]  (* A186499 *)

Table[b[n], {n, 1, 100}]  (* A186500 *)

CROSSREFS

Cf. A186219, A186500, A186511, A186512.

Sequence in context: A136119 A184618 A110882 * A187911 A089230 A171512

Adjacent sequences:  A186496 A186497 A186498 * A186500 A186501 A186502

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 22 2011

STATUS

approved

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Last modified August 1 06:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)