A092543 Table below read by antidiagonals alternately upwards and downwards.

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, 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, 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

Offset: 1

Author

Sam Alexander, Feb 27 2004

Keywords

• nonn
• easy
• tabl

Comments

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

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
• Eric Weisstein's MathWorld, Pairing functions

Crossrefs

Cf. A092542, A056011, A056023.

Programs

Maple

seq(seq(i-abs(i-j),j=1..2*i-1),i=2..20,2); # Robert Israel, Mar 01 2016

Mathematica

Table[ Join[Range[2n], Reverse@Range[2n - 1]], {n, 7}] // Flatten (* Robert G. Wilson v, Sep 28 2006 *)

Formula

T(r,c)=c.

A092542(n)+a(n) = 1+A002024(n). - Enrique Pérez Herrero, Apr 01 2010

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

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