

A213205


T(n,k) = ((k+n)^24*k+3+(1)^k2*(1)^n(k+n)*(1)^(k+n))/2; n , k > 0, read by antidiagonals.


4



1, 5, 4, 2, 3, 6, 10, 9, 14, 13, 7, 8, 11, 12, 15, 19, 18, 23, 22, 27, 26, 16, 17, 20, 21, 24, 25, 28, 32, 31, 36, 35, 40, 39, 44, 43, 29, 30, 33, 34, 37, 38, 41, 42, 45, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 70
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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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(2,1), T(1,2), T(3,1);
. . .
T(1,2*n+1), T(2,2*n), T(2,2*n1), T(1,2*n), ...T(2*n1,3), T(2*n,2), T(2*n,1), T(2*n1,2), T(2*n+1,1);
. . .
Movement along two adjacent antidiagonals  step to the southwest, step to the west, step to the northeast, 2 steps to the south, step to the west and so on. The length of each step is 1.
Table contains:
row 1 accommodates elements A130883 in odd places,
column 3 accommodates elements A130861 in even places;
diagonal 1, located above the main diagonal accommodates elements A033566 in even places,
diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,
diagonal 3, located above the main diagonal accommodates of elements A033585.


LINKS



FORMULA

As table
T(n,k) = ((k+n)^24*k+3+(1)^k2*(1)^n(k+n)*(1)^(k+n))/2.
As linear sequence
a(n) = ((t+2)^24*j+3+(1)^j2*(1)^i(t+2)*(1)^t)/2, where i=nt*(t+1)/2, j=(t*t+3*t+4)/2n, t=floor((1+sqrt(8*n7))/2).


EXAMPLE

The start of the sequence as table:
1....5...2..10...7..19..16...
4....3...9...8..18..17..31...
6...14..11..23..20..36..33...
13..12..22..21..35..34..52...
15..27..24..40..37..57..54...
26..25..39..38..56..55..77...
28..44..41..61..58..82..79...
. . .
The start of the sequence as triangle array read by rows:
1;
5,4;
2,3,6;
10,9,14,13;
7,8,11,12,15;
19,18,23,22,27,26;
16,17,20,21,24,25,28;
. . .
The start of the sequence as array read by rows, the length of row r is 4*r3.
First 2*r2 numbers are from the row number 2*r2 of triangle array, located above.
Last 2*r1 numbers are from the row number 2*r1 of triangle array, located above.
1;
5,4,2,3,6;
10,9,14,13,7,8,11,12,15;
19,18,23,22,27,26,16,17,20,21,24,25,28;
. . .
Row number r contains permutation 4*r3 numbers from 2*r*r5*r+4 to 2*r*rr:
2*r*r5*r+7, 2*r*r5*r+6,...2*r*rr4, 2*r*rr3, 2*r*rr.


MAPLE

T:=(n, k)>((k+n)^24*k+3+(1)^k2*(1)^n(k+n)*(1)^(k+n))/2: seq(seq(T(k, nk), k=1..n1), n=1..13); # Muniru A Asiru, Dec 06 2018


MATHEMATICA

T[n_, k_] := ((n+k)^2  4k + 3 + (1)^k  2(1)^n  (n+k)(1)^(n+k))/2;


PROG

(Python)
t=int((math.sqrt(8*n7)  1)/ 2)
i=nt*(t+1)/2
j=(t*t+3*t+4)/2n
result=((t+2)**24*j+3+(1)**j2*(1)**i(t+2)*(1)**t)/2


CROSSREFS

Cf. A211377, A130883, A100037, A033816, A000384, A091823, A014106, A071355, A130861, A188135, A033567, A033566, A139271, A024847, A033585, A002260, A004736, A003056, A003057.


KEYWORD



AUTHOR



STATUS

approved



