login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A199855 Inverse permutation to A210521. 1
1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79 (list; table; graph; refs; listen; history; text; internal format)
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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
...
T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
T(2,n), T(4,n-2), T(6,n-4), ..., T(n,2),
T(1,n), T(3,n-2), T(5,n-4), ..., T(n-1,2),
T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)} from T(2,2*m-1) to T(2*m,1) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m) to T(2*m,2) length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)} from T(1,2*m) to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
row 1 is alternation of elements A001844 and A084849,
row 2 is alternation of elements A130883 and A058331,
row 3 is alternation of elements A051890 and A096376,
row 4 is alternation of elements A033816 and A005893,
row 6 is alternation of elements A100037 and A093328;
row 5 accommodates elements A097080 in odd places,
row 7 accommodates elements A137882 in odd places,
row 10 accommodates elements A100038 in odd places,
row 14 accommodates elements A100039 in odd places;
column 1 is A093005 and alternation of elements A000384 and A001105,
column 2 is alternation of elements A046092 and A014105,
column 3 is A105638 and alternation of elements A014106 and A056220,
column 4 is alternation of elements A142463 and A014107,
column 5 is alternation of elements A091823 and A054000,
column 6 is alternation of elements A090288 and |A168244|,
column 8 is alternation of elements A059993 and A033537;
column 7 accommodates elements A071355 in odd places,
column 9 accommodates elements |A147973| in even places,
column 10 accommodates elements A139570 in odd places,
column 13 accommodates elements A130861 in odd places.
LINKS
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
FORMULA
T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.
a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).
EXAMPLE
The start of the sequence as table:
1, 4, 5, 11, 13, 22, 25, 37, 41, 56, 61, ...
2, 3, 7, 9, 16, 19, 29, 33, 46, 51, 67, ...
6, 12, 14, 23, 26, 38, 42, 57, 62, 80, 86, ...
8, 10, 17, 20, 30, 34, 47, 52, 68, 74, 93, ...
15, 24, 27, 39, 43, 58, 63, 81, 87, 108, 115, ...
18, 21, 31, 35, 48, 53, 69, 75, 94, 101. 123, ...
28, 40, 44, 59, 64, 82, 88, 109, 116, 140, 148, ...
32, 36, 49, 54, 70, 76, 95, 102, 124, 132, 157, ...
45, 60, 65, 83, 89, 110, 117, 141, 149, 176, 185, ...
50, 55, 71, 77, 96, 103, 125, 133, 158, 167, 195, ...
66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
...
The start of the sequence as triangle array read by rows:
1;
4, 2;
5, 3, 6;
11, 7, 12, 8;
13, 9, 14, 10, 15;
22, 16, 23, 17, 24, 18;
25, 19, 26, 20, 27, 21, 28;
37, 29, 38, 30, 39, 31, 40, 32;
41, 33, 42, 34, 43, 35, 44, 36, 45;
56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
1;
4, 2, 5, 3, 6;
11, 7,12, 8,13, 9,14,10,15;
22,16,23,17,24,18,25,19,26,20,27,21,28;
37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...
PROG
(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4
CROSSREFS
Sequence in context: A267185 A065187 A185511 * A127914 A218035 A090964
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Feb 04 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 06:53 EDT 2024. Contains 370953 sequences. (Running on oeis4.)