1,2

The natural numbers are grouped in chunks of 1, 2, 3, 4,... as (1), (2,3), (4,5,6), (7,8,9,10), etc and each group fills a diagonal in the table. The smallest number in a group is in A000124, the largest in A000217. Numbers in a group are placed on free spots as close as possible to the middle of the diagonal, given preference to the smaller row numbers in the table if there is a draw.

The resulting array is apparently a transposed version of A064789 (if this was also written as an array).

The order of the list table T(n,k):

if n is odd:

T (floor(n+1)/2,floor(n+1)/2), T(floor(n+1)/2-1,floor(n+1)/2+1), T(floor(n+1)/2+1,floor(n+1)/2-1),...T(1,n), T(n,1)

if n is even:

T(floor(n+1)/2-1,floor(n+1)/2+1), T(floor(n+1)/2+1,floor(n+1)/2-1),...T(1,n), T(n,1).

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.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Eric W. Weisstein's World of Mathematics, Pairing functions

a(n) = (i*(i+1) + (j-1)*(j+2*i-4))/2, if j<=i, a(n)=(i*(i+1) + (j-1)*(j+2*i-4))/2 +2*(j-i)-1, if j>i, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2].

The start of the sequence as table:

1....2....5....9...14...20...27 ...

3....4....7...12...18...25...33 ...

6....8...11...16...23...31...40 ...

10..13...17...22...29...38...48 ...

15..19...24...30...37...46...57 ...

21..26...32...39...47...56...67 ...

28..34...41...49...58...68...79 ...

...

The start of the sequence as triangle array read by rows:

1;

2, 3;

5, 4, 6;

9, 7, 8, 10;

14, 12, 11, 13, 15;

20, 18, 16, 17, 19, 21;

27, 25, 23, 22, 24, 26, 28;

. . .

Row number k (k > 1) of the triangle contains a permutation of the set of k numbers from (k^2-k+2)/2, (k^2-k+2)/2 + 1 ,...up to (k^2+k-2)/2 + 1, namely (k^2+k-2)/2, (k^2+k-2)/2 -2,...,(k^2-k+2)/2, (k^2-k+2)/2 + 2,..., (k^2+k-2)/2-1, (k^2+k-2)/2+1.

a[n_] := Module[{i, j, t}, i = n - t(t+1)/2; j = (t^2 + 3t + 4)/2 - n; t = Floor[(-1 + Sqrt[8n - 7])/2]; If[j <= i, (i(i+1) + (j-1)(j + 2i - 4))/2, (i(i+1) + (j-1)(j + 2i - 4))/2 + 2(j-i) - 1]];

Array[a, 68] (* Jean-François Alcover, Nov 21 2018, from Python *)

(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:

m=(i*(i+1) + (j-1)*(j+2*i-4))/2

else:

m=(i*(i+1) + (j-1)*(j+2*i-4))/2 +2*(j-i)-1

Cf. A056011, A056023, A057027, A057028, A064578, A194981, A194982, A188568.

Sequence in context: A279344 A374802 A316669 * A175107 A376685 A085181

Adjacent sequences: A185177 A185178 A185179 * A185181 A185182 A185183

nonn,tabl

Boris Putievskiy, Dec 26 2012

approved