%I #20 Apr 17 2020 11:27:16

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

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

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

%N Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; each k is an R(i(k),j(k)) and a(n)=i(A057027(n))

%C Since A057027 is a permutation of the natural numbers, every natural number occurs infinitely many times in this sequence.

%C Consider the triangle TN := 1; 1, -2; 1, -3, 2; 1, -4, 2, -3; ... Antidiagonal sums give A129819(n+2). TN arises in studying the equation (E) dy/dx=Q(n,x,y)/P(n,x,y) involving saddle-points quantities, P and Q are bidimensional polynomials n=2,3,4.. . (E) leads also for instance to the one-dimension polynomials in A129326, A129587, A130679. - _Paul Curtz_, Aug 16 2008

%C First inverse function (numbers of rows) for pairing function A194982. - _Boris Putievskiy_, Jan 10 2013

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

%F From _Boris Putievskiy_, Jan 10 2013: (Start)

%F a(n) = -((A002260(n)+1)/2)*((-1)^A002260(n)-1)/2+(A004736(n)+A002260(n)/2)*((-1)^A002260(n)+1)/2.

%F a(n) = -((i+1)/2)*((-1)^i-1)/2+(j+i/2)*((-1)^i+1)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2). (End)

%Y Cf. A057059, A194982.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jul 30 2000