OFFSET
1,2
COMMENTS
Self-inverse 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. Natural numbers placed in table T(n,k) by antidiagonals. The order of placement - at the beginning m antidiagonals downwards, next m antidiagonals upwards and so on. T(n,k) read by antidiagonals downwards.
For m = 1 the result is A056011. This sequence is result for m = 2.
A056023 is result for m = 1 and the changed order of placement - at the beginning m antidiagonals upwards, next m antidiagonals downwards and so on.
LINKS
Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
FORMULA
For the general case.
As a table
T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^floor((n + k + m - 2)/m))*n + (1 +(-1)^floor((n + k + m - 2)/m))*k)/2.
As linear sequence
a(n) = ((z - 1)*(z - 2) - (-1 + (-1)^floor((z + m - 2)/m))*i + (1 + (-1)^floor((z + m - 2)/m))*j)/2, where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2), z = i + j.
If we change the order of placement - m antidiagonals upwards, m antidiagonals downwards and so on.
As a table
T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^(floor((n + k + m - 2)/m) + 1))*n + (1 + (-1)^(floor((n + k + m - 2)/m) + 1))*k)/2.
As linear sequence
a(n) = ((z - 1)*(z - 2) - (-1 + (-1)^(floor((z + m - 2)/m) + 1))*i + (1 + (-1)^(floor((z + m - 2)/m) + 1))*j)/2, where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2), z = i + j.
For this sequence.
As a table
T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 +(-1)^floor((n + k)/2))*n + (1 + (-1)^floor((n + k)/2))*k)/2.
As linear sequence
a(n) = ((z - 1)*(z - 2) - (-1 + (-1)^floor(z/2))*i + (1 + (-1)^floor(z/2))*j)/2, where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2), z = i + j.
EXAMPLE
The start of the sequence as table. The direction of the placement denotes by ">" and "v".
v...v v...v
..1...2...6..10..11..16..28..36...
..3...5...9..12..17..27..35..38...
> 4...8..13..18..26..34..39..48...
> 7..14..19..25..33..40..49..63...
.15..20..24..32..41..50..62..74...
.21..23..31..42..51..61..73..84...
>22..30..43..52..60..72..85..98...
>29..44..53..59..71..86..99.113...
. . .
The start of the sequence as triangle array read by rows:
1;
2, 3;
6, 5, 4;
10, 9, 8, 7;
11, 12, 13, 14, 15;
16, 17, 18, 19, 20, 21;
28, 27, 26, 25, 24, 23, 22;
36, 35, 34, 33, 32, 31, 30, 29;
...
Row r consists of r consecutive numbers from r*r/2 - r/2 + 1 to r*r/2 + r.
If r congruent to 1 or 2 mod 4 rows are increasing.
If r congruent to 0 or 3 mod 4 rows are decreasing.
MAPLE
T:=(n, k)->((n+k-1)*(n+k-2)-(-1+(-1)^floor((n+k)/2))*n+(1+(-1)^floor((n+k)/2))*k)/2: seq(seq(T(k, n-k), k=1..n-1), n=1..13); # Muniru A Asiru, Dec 13 2018
MATHEMATICA
T[n_, k_] := ((n+k-1)(n+k-2) - (-1 + (-1)^Floor[(n+k)/2])n + (1 + (-1)^Floor[(n+k)/2]) k)/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 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
z=i+j
result=((z-1)*(z-2)-(-1+(-1)**int(z/2))*i+(1+(-1)**int(z/2))*j)/2
(Maxima) T(n, k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^floor((n + k)/2))*n + (1 +(-1)^floor((n + k)/2))*k)/2$
create_list(T(k, n - k), n, 1, 12, k, 1, n - 1); /* Franck Maminirina Ramaharo, Dec 13 2018 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Feb 19 2013
STATUS
approved