A081344 - OEIS

%I #59 Nov 05 2024 12:17:43

%S 1,2,4,9,3,5,10,8,6,16,25,11,7,15,17,26,24,12,14,18,36,49,27,23,13,19,

%T 35,37,50,48,28,22,20,34,38,64,81,51,47,29,21,33,39,63,65,82,80,52,46,

%U 30,32,40,62,66,100,121,83,79,53,45,31,41,61,67,99,101,122,120,84,78,54

%N Natural numbers in square maze arrangement, read by antidiagonals.

%C Arrange the natural numbers by taking clockwise and counterclockwise turns. Begin (LL) and then repeat (RRR)(LLL).

%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. - _Boris Putievskiy_, Dec 16 2012

%C For generalizations see A219159, A213928. - _Boris Putievskiy_, Mar 10 2013

%H Boris Putievskiy, <a href="/A081344/b081344.txt">Rows n = 1..100 of triangle, flattened</a>

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations 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 From _Boris Putievskiy_, Dec 19 2012: (Start)

%F a(n) = (i-1)^2 + i + (i-j)*(-1)^(i-1) if i >= j,

%F a(n) = (j-1)^2 + j - (j-i)*(-1)^(j-1) if i <  j,

%F where

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

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

%F t = floor((-1 + sqrt(8*n-7))/2). (End)

%e The start of the sequence as table T(i,j), i,j > 0:

%e 1 .. 4 .. 5 .. 16...

%e 2 .. 3 .. 6 .. 15...

%e 9 .. 8 .. 7 .. 14...

%e 10..11 ..12 .. 13...

%e . . .

%e Enumeration by boustrophedonic ("ox-plowing") method: If i >= j: T(i,i)=(i-1)^2+i + (i-j)*(-1)^(i-1), if i  < j: T(i,j)=(j-1)^2+j - (j-i)*(-1)^(j-1). - _Boris Putievskiy_, Dec 19 2012

%t T[n_, k_] := T[n, k] = Which[OddQ[n] && k==1, n^2, EvenQ[k] && n==1, k^2, EvenQ[n] && k==1, T[n-1, 1]+1, OddQ[k] && n==1, T[1, k-1]+1, k <= n, T[n, k-1]+1 - 2 Mod[n, 2], True, T[n-1, k]-1 + 2 Mod[k, 2]]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 20 2019 *)

%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 if j >= i:

%o      m=(j-1)**2 + j + (j-i)*(-1)**(j-1)

%o else:

%o      m=(i-1)**2 + i - (i-j)*(-1)**(i-1)

%o # _Boris Putievskiy_, Dec 19 2012

%o (Python)

%o from math import isqrt

%o def A081344(n):

%o     t = (k:=isqrt(m:=n<<1))+((m<<2)>(k<<2)*(k+1)+1)-1

%o     i, j = n-(t*(t+1)>>1), (t*(t+3)>>1)+2-n

%o     r = max(i,j)

%o     return (r-1)**2+r+(j-i if r&1 else i-j) # _Chai Wah Wu_, Nov 04 2024

%Y Cf. A219159, A213928. The main diagonal is A002061. The following appear within interlaced sequences: A016754, A001844, A053755, A004120. The first row is A081345. The first column is A081346. The inverse permutation A194280, the first inverse function (numbers of rows) A220603, the second inverse function (numbers of columns) A220604.

%K nonn,tabl

%O 1,2

%A _Paul Barry_, Mar 19 2003