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Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 9 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=9.

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 10 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=10.

Let A be sequence A092542 (this sequence) and B be sequence A092543 (1, 2, 1, 1, 2, 3, 4, ...). Under upper trimming or lower trimming, A transforms into B and B transforms into A. Also, B gives the number of times each element of A appears. For example, A(7) = 1 and B(7) = 4 because the 1 in A(7) is the fourth 1 to appear in A. - Kerry Mitchell, Dec 28 2005

First inverse function (numbers of rows) for pairing function A056023 and second inverse function (numbers of columns) for pairing function A056011. - Boris Putievskiy, Dec 24 2012

The rational numbers a(n)/A092543(n) can be systematically ordered and numbered in this way, as Georg Cantor first proved in 1873. - Martin Renner, Jun 05 2016

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.

Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This table read layer by layer clockwise is A194280. This table read by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - Boris Putievskiy, Mar 14 2013

1 2 3 4 5 ...

1 2 3 4 5 ...

1 2 3 4 5 ...

1 2 3 4 5 ...

...

Let A be sequence A092543 (this sequence) and B be sequence A092542 (1, 1, 2, 3, 2, 1, 1, ...). Under upper trimming or lower trimming, A transforms into B and B transforms into A. Also, B gives the number of times each element of A appears. For example, A(4) = 1 and B(4) = 3 because the 1 in A(4) is the third 1 to appear in A. - Kerry Mitchell, Dec 28 2005

First inverse function (numbers of rows) for pairing function A056011 and second inverse function (numbers of columns) for pairing function A056023. - Boris Putievskiy, Dec 24 2012

The rational numbers A092542(n)/a(n) can be systematically ordered and numbered in this way, as Georg Cantor first proved in 1873. - Martin Renner, Jun 05 2016

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.

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.

1

3 2

4 5 6

10 9 8 7

11 12 13 14 15

21 20 19 18 17 16

22 23 24 25 26 27 28

36 35 34 33 32 31 30 29

37 38 39 40 41 42 43 44 45

55 54 53 52 51 50 49 48 47 46

56 57 58 59 60 61 62 63 64 65 66

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 b(z) be a sequence of integers and denote number of antidiagonal table T(n,k) by z=n+k-1. Natural numbers placed in table T(n,k) by antidiagonals. The order of placement - by antidiagonal downwards, if b(z) is odd; by antidiagonal upwards, if b(z) is even. T(n,k) read by antidiagonals downwards. For A218890 -- the order of placement -- at the beginning m antidiagonals downwards, next m antidiagonals upwards and so on - b(z)=floor((z+m-1)/m). For this sequence b(z)=z^2 mod 3. (This comment should be edited for clarity, Joerg Arndt, Dec 11 2014)