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.

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

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