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A213195 Second inverse function (of columns) for pairing function A211377. 1

%I #23 Nov 29 2023 10:37:52

%S 1,3,2,1,2,1,5,4,3,4,3,2,1,2,1,7,6,5,6,5,4,3,4,3,2,1,2,1,9,8,7,8,7,6,

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

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

%N Second inverse function (of columns) for pairing function A211377.

%H Boris Putievskiy, <a href="/A213195/b213195.txt">Rows n = 1..140 of triangle, flattened</a>

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F See Python program.

%e The start of the sequence as triangle array read by rows:

%e 1;

%e 3,2;

%e 1,2,1;

%e 5,4,3,4;

%e 3,2,1,2,1;

%e 7,6,5,6,5,4;

%e 3,4,3,2,1,2,1;

%e . . .

%e The start of the sequence as array read by rows, the length of row r is 4*r-3.

%e First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.

%e Last 2*r-1 numbers are from the row number 2*r-1 of above triangle array.

%e 1;

%e 3,2,1,2,1;

%e 5,4,3,4,3,2,1,2,1;

%e 7,6,5,6,5,4,3,4,3,2,1,2,1;

%e ...

%e Row number r is 2*r-1, 2*r-2, 2*r-3, 2*r-2, {row number r-1}.

%o (Python)

%o t=int((math.sqrt(8*n-7) - 1)/ 2)

%o i=n-t*(t+1)/2

%o j=(t*t+3*t+4)/2-n

%o result=((1+(-1)**i)*((1+(-1)**j)*2*int((j+2)/4)-(-1+(-1)**j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)**i)*((1+(-1)**j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)**j)*(1+2*int(j/4))))/4

%Y Cf. A211377.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Mar 01 2013

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