A186352 - OEIS

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A186352

Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the odd numbers and the triangular numbers. Complement of A186353.

2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..120.

FORMULA

a(n)=n+floor(-1/2+sqrt(4n-3/4))=A186352(n).

b(n)=n+floor((n^2+n+1)/4)=A186353(n).

EXAMPLE

First, write

1..3..5..7..9..11..13..15..17..21..23.. (odds)

1..3....6.....10.......15......21.... (triangular)

Then replace each number by its rank, where ties are settled by ranking the odd number after the triangular:

a=(2,4,5,7,8,10,11,13,14,15,....)=A186352

b=(1,3,6,9,12,16,21,26,31,37,...)=A186353.

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=2; v=-1; x=1/2; y=1/2; (* odds and triangular *)

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

a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *)

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

b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *)

Table[a[n], {n, 1, 120}] (* A186352 *)

Table[b[n], {n, 1, 100}] (* A186353 *)

CROSSREFS

Cf. A186350, A186351, A186353.

Sequence in context: A078934 A026496 A056127 * A186157 A072633 A037087

Adjacent sequences: A186349 A186350 A186351 * A186353 A186354 A186355

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 18 2011

STATUS

approved