cp's OEIS Frontend

A081344 Natural numbers in square maze arrangement, read by antidiagonals.

%I A081344 #72 Jun 11 2025 03:57:17
%S A081344 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 A081344 35,37,50,48,28,22,20,34,38,64,81,51,47,29,21,33,39,63,65,82,80,52,46,
%U A081344 30,32,40,62,66,100,121,83,79,53,45,31,41,61,67,99,101,122,120,84,78,54
%N A081344 Natural numbers in square maze arrangement, read by antidiagonals.
%C A081344 Arrange the natural numbers by taking clockwise and counterclockwise turns. Begin (LL) and then repeat (RRR)(LLL).
%C A081344 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 A081344 For generalizations see A219159, A213928. - _Boris Putievskiy_, Mar 10 2013
%H A081344 Boris Putievskiy, <a href="/A081344/b081344.txt">Rows n = 1..100 of triangle, flattened</a>
%H A081344 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%H A081344 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A081344 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A081344 From _Boris Putievskiy_, Dec 19 2012: (Start)
%F A081344 a(n) = (i-1)^2 + i + (i-j)*(-1)^(i-1) if i >= j,
%F A081344 a(n) = (j-1)^2 + j - (j-i)*(-1)^(j-1) if i <  j,
%F A081344 where
%F A081344 i = n - t*(t+1)/2,
%F A081344 j = (t*t + 3*t + 4)/2-n,
%F A081344 t = floor((-1 + sqrt(8*n-7))/2). (End)
%F A081344 Enumeration by boustrophedonic ("ox-plowing") method: If i >= j: T(i,j)=(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
%F A081344 T(i,j) = m^2 - m + 1 - (i - j)*(-1)^m where m = max(i,j). - _Ziad Ahmed_, Jun 09 2025
%e A081344 The start of the sequence as table T(i,j), i,j > 0:
%e A081344    1   4    5    16 ...
%e A081344    2   3    6    15 ...
%e A081344    9   8    7    14 ...
%e A081344   10  11   12    13 ...
%e A081344   ....
%t A081344 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 A081344 (Python)
%o A081344 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A081344 i=n-t*(t+1)/2
%o A081344 j=(t*t+3*t+4)/2-n
%o A081344 if j >= i:
%o A081344      m=(j-1)**2 + j + (j-i)*(-1)**(j-1)
%o A081344 else:
%o A081344      m=(i-1)**2 + i - (i-j)*(-1)**(i-1)
%o A081344 # _Boris Putievskiy_, Dec 19 2012
%o A081344 (Python)
%o A081344 from math import isqrt
%o A081344 def A081344(n):
%o A081344     t = (k:=isqrt(m:=n<<1))+((m<<2)>(k<<2)*(k+1)+1)-1
%o A081344     i, j = n-(t*(t+1)>>1), (t*(t+3)>>1)+2-n
%o A081344     r = max(i,j)
%o A081344     return (r-1)**2+r+(j-i if r&1 else i-j) # _Chai Wah Wu_, Nov 04 2024
%Y A081344 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 A081344 nonn,tabl
%O A081344 1,2
%A A081344 _Paul Barry_, Mar 19 2003