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A210535 Second inverse function (numbers of columns) for pairing function A209293.

%I A210535 #23 Feb 16 2025 08:33:17
%S A210535 1,2,1,2,3,1,2,4,3,1,2,4,5,3,1,2,4,6,5,3,1,2,4,6,7,5,3,1,2,4,6,8,7,5,
%T A210535 3,1,2,4,6,8,9,7,5,3,1,2,4,6,8,10,9,7,5,3,1,2,4,6,8,10,11,9,7,5,3,1,2,
%U A210535 4,6,8,10,12,11,9,7,5,3,1,2,4,6,8,10,12
%N A210535 Second inverse function (numbers of columns) for pairing function A209293.
%H A210535 Boris Putievskiy, <a href="/A210535/b210535.txt">Rows n = 1..140 of triangle, flattened</a>
%H A210535 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 A210535 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%F A210535 a(n) = 2*A200260(n)-A101688(n)*(4*A002260(n)-2*A003056(n)-3).
%F A210535 a(n) = 2*i-v*(4*i-2*t-3), where t = floor((-1+sqrt(8*n-7))/2), i = n-t*(t+1)/2, v = floor((2*n+1-t*(t+1))/(t+3)).
%e A210535 The start of the sequence as triangle array read by rows:
%e A210535   1;
%e A210535   2,1;
%e A210535   2,3,1;
%e A210535   2,4,3,1;
%e A210535   2,4,5,3,1;
%e A210535   2,4,6,5,3,1;
%e A210535   2,4,6,7,5,3,1;
%e A210535   2,4,6,8,7,5,3,1;
%e A210535   . . .
%e A210535 Row number r contains permutation numbers from 1 to r: 2,4,6,...2*floor(r/2),2*floor(r/2)-1,2*floor(r/2)-3,...3,1.
%o A210535 (Python)
%o A210535 t=int((math.sqrt(8*n-7)-1)/2)
%o A210535 i=n-t*(t+1)/2
%o A210535 v=int((2*n+1-t*(t+1))/(t+3))
%o A210535 result=2*i-v*(4*i-2*t-3)
%Y A210535 Cf. A209293, A200260, A101688, A003056, A220073.
%K A210535 nonn
%O A210535 1,2
%A A210535 _Boris Putievskiy_, Jan 28 2013