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A219159
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Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 2 layers clockwise and so on. T(n,k) read by antidiagonals.
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3
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1, 4, 2, 5, 3, 9, 10, 6, 8, 16, 25, 11, 7, 15, 17, 36, 24, 12, 14, 18, 26, 37, 35, 23, 13, 19, 27, 49, 50, 38, 34, 22, 20, 28, 48, 64, 81, 51, 39, 33, 21, 29, 47, 63, 65, 100, 80, 52, 40, 32, 30, 46, 62, 66, 82, 101, 99, 79, 53, 41, 31, 45, 61, 67, 83, 121
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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.
In general, let m be natural number. Layer is pair of sides of square table T(n,k) from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Natural numbers placed in the table T(n,k) layer by layer. The order of placement - at the beginning m layers counterclockwise, next m layers clockwise and so on. T(n,k) read by antidiagonals.
For m = 1 the result is A081344. This sequence is result for m = 2.
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LINKS
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FORMULA
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For general case.
As table
T(n,k) = ((1 + (-1)^floor((k - 1)/m))*(k^2 - n + 1) - (-1 + (-1)^floor((k - 1)/m))*((k - 1)^2 + n))/2, if k >= n; T(n,k) = ((1 + (-1)^(floor((n - 1)/m) + 1))*(n^2 - k + 1) - (-1 + (-1)^(floor((n - 1)/m) + 1))*((n - 1)^2 +k))/2, if n > k.
As linear sequence
a(n) = ((1 + (-1)^floor((j - 1)/m))*(j^2 - i + 1) - (-1 + (-1)^floor((j - 1)/m))*((j - 1)^2 + i))/2, if j >= i; a(n) = ((1 + (-1)^(floor((i - 1)/m) + 1))*(i^2 - j + 1) - (-1 + (-1)^(floor((i - 1)/m) + 1))*((i - 1)^2 + j))/2, if i > j; where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2).
For this sequence.
As table
T(n,k) = ((1 + (-1)^floor((k - 1)/2))*(k^2 - n + 1) - (-1 + (-1)^floor((k - 1)/2))*((k - 1)^2 + n))/2, if k >= n; T(n,k) = ((1 + (-1)^(floor((n - 1)/2) + 1))*(n^2 - k + 1) - (-1 + (-1)^(floor((n - 1)/2) + 1))*((n - 1)^2 + k))/2, if n > k.
As linear sequence
a(n) = ((1 + (-1)^floor((j - 1)/2))*(j^2 - i + 1) - (-1 + (-1)^floor((j - 1)/2))*((j - 1)^2 + i))/2, if j >= i; a(n) = ((1 + (-1)^(floor((i - 1)/2) + 1))*(i^2 - j + 1) - (-1 + (-1)^(floor((i - 1)/2) + 1))*((i - 1)^2 + j))/2, if i > j; where i = n - t*(t + 1)/2, j = (t*t +3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2).
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EXAMPLE
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The start of the sequence as table.
The direction of the placement denotes by ">" and "v".
v..v v...v
.>1...4...5..10..25..36..37..50...
.>2...3...6..11..24..35..38..51...
..9...8...7..12..23..34..39..52...
.16..15..14..13..22..33..40..53...
>17..18..19..20..21..32..41..54...
>26..27..28..29..30..31..42..55...
.49..48..47..46..45..44..43..56...
.64..63..62..61..60..59..58..57...
. . .
The start of the sequence as triangle array read by rows:
1;
4, 2;
5, 3, 9;
10, 6, 8, 16;
25, 11, 7, 15, 17;
36, 24, 12, 14, 18, 26;
37, 35, 23, 13, 19, 27, 49;
50, 38, 34, 22, 20, 28, 48, 64;
...
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MATHEMATICA
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T[n_, k_] := If[k >= n, ((1 + (-1)^Floor[(k-1)/2])(k^2 - n + 1) - (-1 + (-1)^Floor[(k-1)/2])((k-1)^2 + n))/2, ((1 + (-1)^(Floor[(n-1)/2] + 1))(n^2 - k + 1) - (-1 + (-1)^(Floor[(n-1)/2] + 1))((n-1)^2 + k))/2];
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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
if j >= i:
result=((1+(-1)**int((j-1)/2))*(j**2-i+1)-(-1+(-1)**int((j-1)/2))*((j-1)**2 +i))/2
else:
result=((1+(-1)**(int((i-1)/2)+1))*(i**2-j+1)-(-1+(-1)**(int((i-1)/2)+1))*((i-1)**2 +j))/2
(Maxima) T(n, k) := if k >= n then ((1 + (-1)^floor((k - 1)/2))*(k^2 - n + 1) - (-1 + (-1)^floor((k - 1)/2))*((k - 1)^2 + n))/2 else ((1 + (-1)^(floor((n - 1)/2) + 1))*(n^2 - k + 1) - (-1 + (-1)^(floor((n - 1)/2) + 1))*((n - 1)^2 +k))/2$
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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