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%I #34 Feb 15 2022 13:00:37
%S 1,2,1,1,2,3,4,3,2,1,1,2,3,4,5,6,5,4,3,2,1,1,2,3,4,5,6,7,8,7,6,5,4,3,
%T 2,1,1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,11,
%U 12,11,10,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,11,12,13,14,13,12,11,10,9,8
%N Table below read by antidiagonals alternately upwards and downwards.
%C 1 2 3 4 5 ...
%C 1 2 3 4 5 ...
%C 1 2 3 4 5 ...
%C 1 2 3 4 5 ...
%C ...
%C 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
%C 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
%C 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
%D Amir D. Aczel, "The Mystery of the Aleph, Mathematics, the Kabbalah and the Search for Infinity", Barnes & Noble, NY 2000, page 112.
%H Robert Israel, <a href="/A092543/b092543.txt">Table of n, a(n) for n = 1..10000</a>
%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 T(r,c)=c.
%F A092542(n)+a(n) = 1+A002024(n). - _Enrique Pérez Herrero_, Apr 01 2010
%F a(n) = ((-1)^t+1)*i/2-((-1)^t-1)*j)/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
%F G.f.: (1-x)^(-2)*Sum_{i>=0} x^(2*i^2+i+1)*(1-x^(2*i+2))^2. - _Robert Israel_, Mar 01 2016
%p seq(seq(i-abs(i-j),j=1..2*i-1),i=2..20,2); # _Robert Israel_, Mar 01 2016
%t Table[ Join[Range[2n], Reverse@Range[2n - 1]], {n, 7}] // Flatten (* _Robert G. Wilson v_, Sep 28 2006 *)
%Y Cf. A092542, A056011, A056023.
%K nonn,easy,tabl
%O 1,2
%A _Sam Alexander_, Feb 27 2004
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