A092542 - OEIS

%I #40 Feb 16 2025 08:32:52

%S 1,1,2,3,2,1,1,2,3,4,5,4,3,2,1,1,2,3,4,5,6,7,6,5,4,3,2,1,1,2,3,4,5,6,

%T 7,8,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1,1,

%U 2,3,4,5,6,7,8,9,10,11,12,13,12,11,10,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9

%N Table whose n-th row is constant and equal to n, read by antidiagonals alternately upwards and downwards.

%C 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

%C 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

%C 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

%D Amir D. Aczel, "The Mystery of the Aleph, Mathematics, the Kabbalah and the Search for Infinity", Barnes & Noble, NY 2000, page 112.

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>

%F a(n) = ((-1)^t+1)*j/2-((-1)^t-1)*i/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 24 2012

%e The table

%e   1 1 1 1 1 ...

%e   2 2 2 2 2 ...

%e   3 3 3 3 3 ...

%e   4 4 4 4 4 ...

%e gives

%e   1;

%e   1 2;

%e   3 2 1;

%e   1 2 3 4;

%e   5 4 3 2 1;

%e   1 2 3 4 5 6;

%t Table[ Join[Range[2n - 1], Reverse@ Range[2n - 2]], {n, 8}] // Flatten (* _Robert G. Wilson v_, Sep 28 2006 *)

%Y Cf. A092543, A056011, A056023.

%Y Variants of Cantor's enumeration are: A352911, A366191, A319571, A354266.

%K easy,nonn,tabl

%O 1,3

%A _Sam Alexander_, Feb 27 2004

%E Name edited by _Michel Marcus_, Dec 14 2023