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A186346 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2.  Complement of A186347. 4
3, 5, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See A186350 for a discussion of adjusted joint rank sequences.

LINKS

Table of n, a(n) for n=1..112.

FORMULA

a(n)=n+floor(sqrt(8n-1/2))=A186346(n).

b(n)=n+floor((n^2+1/2)/8)=A186347(n).

EXAMPLE

First, write

....8....16..24..32..40..48..56..64..72..80.. (8i)

1..4..9..16...25...36......49....64.......81 (squares)

Then replace each number by its rank, where ties are settled by ranking 8i before the square:

a=(3,5,7,9,11,12,14,15,17,..)=A186346

b=(1,2,4,6,8,10,13,16,19,...)=A186347.

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=8; v=0; x=1; y=0;

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

a[n_]:=n+Floor[h[n]];

k[n_]:=(x*n^2+y*n-v+d)/u;

b[n_]:=n+Floor[k[n]];

Table[a[n], {n, 1, 120}]  (* A186346 *)

Table[b[n], {n, 1, 100}]  (* A186347 *)

CROSSREFS

Cf. A186350, A186347, A186348, A186349.

Sequence in context: A275669 A191275 A260392 * A248633 A137228 A186288

Adjacent sequences:  A186343 A186344 A186345 * A186347 A186348 A186349

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 20 2011

STATUS

approved

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)