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A220604 Second inverse function (numbers of columns) for pairing function A081344.

%I A220604 #26 Feb 15 2022 12:56:04
%S A220604 1,1,2,2,3,3,3,2,1,1,2,3,4,4,4,4,5,5,5,5,5,4,3,2,1,1,2,3,4,5,6,6,6,6,
%T A220604 6,6,7,7,7,7,7,7,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,8,8,8,8,8,8,8,9,9,9,9,
%U A220604 9,9,9,9,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,10,10,10,10,10,10,10,10,10
%N A220604 Second inverse function (numbers of columns) for pairing function A081344.
%H A220604 Boris Putievskiy, <a href="/A220604/b220604.txt">Rows n = 1..140 of triangle, flattened</a>
%H A220604 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A220604 As a linear array, the sequence is a(n) = mod(t;2)*min{t; t^2 - n + 1} + mod(t + 1; 2)*min{t; n - (t - 1)^2}, where t=floor[sqrt(n-1)]+1.
%e A220604 The start of the sequence as triangle array T(n,k) read by rows, row number k contains 2k-1 numbers:
%e A220604   1;
%e A220604   1,2,2;
%e A220604   3,3,3,2,1;
%e A220604   1,2,3,4,4,4,4;
%e A220604   ...
%e A220604 If k is even the row is 1,2,...,k,k...k (k times repetition "k" at the end of row).
%e A220604 If k is odd the row is k,k,...k,k-1,k-2,...1 (k times repetition "k" at the start of row).
%o A220604 (Python)
%o A220604 t=int(math.sqrt(n-1))+1
%o A220604 j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
%Y A220604 Cf. A081344.
%K A220604 nonn,tabf
%O A220604 1,3
%A A220604 _Boris Putievskiy_, Dec 17 2012