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A219159 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. 3
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 (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.
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.
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
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).
EXAMPLE
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;
...
MATHEMATICA
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];
Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 11 2018 *)
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
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$
create_list(T(k, n - k), n, 1, 12, k, 1, n - 1); /* Franck Maminirina Ramaharo, Dec 11 2018 */
CROSSREFS
Sequence in context: A127914 A218035 A090964 * A213928 A065189 A165275
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Feb 19 2013
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)