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A211394
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T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.
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4
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1, 5, 6, 2, 3, 4, 12, 13, 14, 15, 7, 8, 9, 10, 11, 23, 24, 25, 26, 27, 28, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 29, 30, 31, 32, 33, 34, 35, 36, 37, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 80
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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(1,3), T(2,2), T(3,1);
T(1,2), T(2,1);
. . .
T(1,n), T(2,n-1), T(3,n-2), ... T(n,1);
T(1,n-1), T(2,n-3), T(3,n-4),...T(n-1,1);
. . .
First row matches with the elements antidiagonal {T(1,n), ... T(n,1)},
second row matches with the elements antidiagonal {T(1,n-1), ... T(n-1,1)}.
Table contains:
row 2 accommodates elements A033816 in even places,
row 3 accommodates elements A100037 in odd places,
row 5 accommodates elements A100038 in odd places;
column 5 accommodates elements A033537 in even places,
column 9 accommodates elements A100041 in even places;
diagonal 1, located above the main diagonal is A001844,
diagonal 2, located above the main diagonal is A001105,
diagonal 3, located above the main diagonal is A046092,
diagonal 4, located above the main diagonal is A056220,
diagonal 5, located above the main diagonal is A142463,
diagonal 6, located above the main diagonal is A054000,
diagonal 7, located above the main diagonal is A090288,
diagonal 9, located above the main diagonal is A059993,
diagonal 10, located above the main diagonal is |A147973|,
diagonal 11, located above the main diagonal is A139570;
diagonal 1, located under the main diagonal is A051890,
diagonal 2, located under the main diagonal is A005893,
diagonal 3, located under the main diagonal is A097080,
diagonal 4, located under the main diagonal is A093328,
diagonal 5, located under the main diagonal is A137882.
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LINKS
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FORMULA
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T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2.
As linear sequence
a(n) = (t+2)*(t+1)/2 - (t+1)*(-1)^t-j+2, where j=(t*t+3*t+4)/2-n and 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....5...2..12...7..23..16...
6....3..13...8..24..17..39...
4...14...9..25..18..40..31...
15..10..26..19..41..32..60...
11..27..20..42..33..61..50...
28..21..43..34..62..51..85...
22..44..35..63..52..86..73...
. . .
The start of the sequence as triangle array read by rows:
1;
5,6;
2,3,4;
12,13,14,15;
7,8,9,10,11;
23,24,25,26,27,28;
16,17,18,19,20,21,22;
. . .
Row number r matches with r numbers segment {(r+1)*r/2-r*(-1)^(r+1)-r+2,... (r+1)*r/2-r*(-1)^(r+1)+1}.
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MATHEMATICA
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T[n_, k_] := (n+k)(n+k-1)/2 - (-1)^(n+k)(n+k-1) - k + 2;
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PROG
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(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
j=(t*t+3*t+4)/2-n
result=(t+2)*(t+1)/2-(t+1)*(-1)**t-j+2
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CROSSREFS
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Cf. A130883, A096376, A033816, A100037, A100038, A084849, A000384, A014106, A014105, A014107, A091823, A071355, A168244, A033537, A100040, A130861, A100041, A058331, A001844, A001105, A046092, A056220, A142463, A054000, A090288, A059993, A147973, A139570, A051890, A005893, A097080, A093328, A137882.
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KEYWORD
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AUTHOR
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STATUS
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approved
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