A216253 - OEIS

%I #17 Nov 29 2023 08:10:35

%S 1,2,5,4,3,7,10,8,6,12,14,23,20,17,9,11,13,16,26,38,43,39,21,24,15,18,

%T 27,31,35,48,63,58,42,30,25,22,19,29,34,57,53,69,76,70,64,49,36,32,28,

%U 40,44,59,54,82,88,109,102,95,75,81,52,47,33,37,41,46,62

%N A213196 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

%C Permutation of the natural numbers.

%C a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

%C Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method.

%C Let m be natural number. The order of the list:

%C   T(1,1)=1;

%C   T(2,1), T(2,2), T(1,2);

%C   . . .

%C   T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);

%C   T(2*m,1),   T(2*m,2),   ... T(2*m,2*m-1), T(2*m,2*m),     T(2*m-1,2*m), ... T(1,2*m);

%C . . .

%C The first row is layer read clockwise, the second row is layer counterclockwise.

%H Boris Putievskiy, <a href="/A216253/b216253.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 a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where m1=(3*i+j-1-(-1)^i+(i+j-2)*(-1)^(i+j))/4, m2=((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, i=(t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j=(t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.

%e The start of the sequence as table:

%e   1....4...3..11..13...

%e   2....5...7...9..16...

%e   6....8..10..17..26...

%e   12..14..23..20..38...

%e   15..24..21..39..43...

%e   . . .

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

%e   1;

%e   2,5,4;

%e   3,7,10,8,6;

%e   12,14,23,20,17,9,11;

%e   13,16,26,38,43,39,21,24,15;

%e   . . .

%e Row number r contains 2*r-1 numbers.

%o (Python)

%o t=int((math.sqrt(n-1)))+1

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

%o j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)

%o m1=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4

%o m2=((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

%o m=(m1+m2-1)*(m1+m2-2)/2+m1

%Y Cf. A213196, A081344, A211377, A214929.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Mar 15 2013