

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,
row 2 is alternation of elements A100037 and A033816;
column 1 is alternation of elements A000384 and A091823,
column 2 is alternation of elements A014106 and A071355,
column 3 accommodates elements A130861 in even places;
main diagonal is alternation of elements A188135 and A033567,
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

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 Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers


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) = (A003057(n)^24*A004736(n)+3+(1)^A004736(n)2*(1)^A002260(n)A003057(n)*(1)^A003056(n))/2;
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;
Table[T[nk+1, k], {n, 1, 12}, {k, n, 1, 1}] // Flatten (* JeanFrançois Alcover, Dec 06 2018 *)


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.
Sequence in context: A074825 A225063 A309442 * A094778 A260849 A246746
Adjacent sequences: A213202 A213203 A213204 * A213206 A213207 A213208


KEYWORD

nonn,tabl


AUTHOR

Boris Putievskiy, Feb 15 2013


STATUS

approved



