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A199855
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Inverse permutation of A210521.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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 integer number, m>0.
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.
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LINKS
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Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO]
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers
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FORMULA
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As table
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.
As linear sequence
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=int((math.sqrt(8*n-7) - 1)/ 2).
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EXAMPLE
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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.
. . .
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PROG
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(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
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CROSSREFS
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Cf. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.
Sequence in context: A267185 A065187 A185511 * A127914 A218035 A090964
Adjacent sequences: A199852 A199853 A199854 * A199856 A199857 A199858
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KEYWORD
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nonn,tabl
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AUTHOR
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Boris Putievskiy, Feb 04 2013
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STATUS
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approved
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