Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #35 Dec 14 2023 08:57:16
%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="http://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