A209268 Inverse permutation A054582.

1, 2, 3, 4, 6, 5, 10, 7, 15, 9, 21, 8, 28, 14, 36, 11, 45, 20, 55, 13, 66, 27, 78, 12, 91, 35, 105, 19, 120, 44, 136, 16, 153, 54, 171, 26, 190, 65, 210, 18, 231, 77, 253, 34, 276, 90, 300, 17, 325, 104, 351, 43, 378, 119, 406, 25, 435, 135, 465, 53, 496, 152

Offset

1,2

Comments

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Links

Boris Putievskiy, Table of n, a(n) for n = 1..10000

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

R. J. Mathar, oeisPy

Eric W. Weisstein, MathWorld: Pairing functions

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = (((A003602)+A007814(n))^2 - A007814(n) + A003602(n))/2.

a(n) = ((x+y)^2-x+y)/2, where x = max {k: 2^k | n}, y = (n+2^x)/2^(x+1).

Example

The start of the sequence for n = 1..32 as table, distributed by exponent of highest power of 2 dividing n:
   |   Exponent of highest power of 2 dividing n
n  |--------------------------------------------------
   |    0      1      2       3      4         5    ...
------------------------------------------------------
1  |....1
2  |...........2
3  |....3
4  |..................4
5  |....6
6  |...........5
7  |...10
8  |..........................7
9  |...15
10 |...........9
11 |...21
12 |..................8
13 |...28
14 |..........14
15 |...36
16 |................................11
17 |...45
18 |..........20
19 |...55
20 |.................13
21 |...66
22 |..........27
23 |...78
24 |................................12
25 |...91
26 |..........35
27 |..105
28 |.................19
29 |..120
30 |..........44
31 |..136
32 |.........................................16
. . .
Let r_c be number row inside the column number c.
r_c = (n+2^c)/2^(c+1).
The column number 0 contains numbers r_0*(r_0+1)/2,     A000217,
The column number 1 contains numbers r_1*(r_1+3)/2,     A000096,
The column number 2 contains numbers r_2*(r_2+5)/2 + 1, A034856,
The column number 3 contains numbers r_3*(r_3+7)/2 + 3, A055998,
The column number 4 contains numbers r_4*(r_4+9)/2 + 6, A046691.

Mathematica

a[n_] := (v = IntegerExponent[n, 2]; (1/2)*(((1/2)*(n/2^v + 1) + v)^2 + (1/2)*(n/2^v + 1) - v)); Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jan 15 2013, from 1st formula *)

Prog

(Python)
f = open("result.csv", "w")
def A007814(n):
### author        Richard J. Mathar 2010-09-06 (Start)
### http://oeis.org/wiki/User:R._J._Mathar/oeisPy/oeisPy/oeis_bulk.py
        a = 0
        nshft = n
        while (nshft %2 == 0):
                a += 1
                nshft >>= 1
        return a
###(End)
for  n in range(1, 10001):
     x = A007814(n)
     y = (n+2**x)/2**(x+1)
     m = ((x+y)**2-x+y)/2
     f.write('%d; %d; %d; %d; \n' % (n, x, y, m))
f.close()

Crossrefs

Cf. A054582, A003602, A007814, A000217, A000096, A034856, A055998, A046691, A014132.

Sequence in context: A191545 A064275 A080998 * A257798 A183079 A119629

Adjacent sequences: A209265 A209266 A209267 * A209269 A209270 A209271

Keyword

nonn

Author

Boris Putievskiy, Jan 15 2013

Status

approved