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

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, 35, 37, 50, 48, 28, 22, 20, 34, 38, 64, 81, 51, 47, 29, 21, 33, 39, 63, 65, 82, 80, 52, 46, 30, 32, 40, 62, 66, 100, 121, 83, 79, 53, 45, 31, 41, 61, 67, 99, 101, 122, 120, 84, 78, 54

Offset

1,2

Comments

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

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

For generalizations see A219159, A213928. - Boris Putievskiy, Mar 10 2013

Links

Boris Putievskiy, Rows n = 1..100 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Eric Weisstein's World of Mathematics, Pairing functions

Index entries for sequences that are permutations of the natural numbers

Formula

From Boris Putievskiy, Dec 19 2012: (Start)

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

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

where

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

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

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

Example

The start of the sequence as table T(i,j), i,j > 0:
1 .. 4 .. 5 .. 16...
2 .. 3 .. 6 .. 15...
9 .. 8 .. 7 .. 14...
10..11 ..12 .. 13...
. . .
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

Mathematica

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 *)

Prog

(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j >= i:
     m=(j-1)**2 + j + (j-i)*(-1)**(j-1)
else:
     m=(i-1)**2 + i - (i-j)*(-1)**(i-1)
# Boris Putievskiy, Dec 19 2012
(Python)
from math import isqrt
def A081344(n):
    t = (k:=isqrt(m:=n<<1))+((m<<2)>(k<<2)*(k+1)+1)-1
    i, j = n-(t*(t+1)>>1), (t*(t+3)>>1)+2-n
    r = max(i, j)
    return (r-1)**2+r+(j-i if r&1 else i-j) # Chai Wah Wu, Nov 04 2024

Crossrefs

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.

Sequence in context: A155523 A019912 A354903 * A227272 A021405 A201946

Adjacent sequences: A081341 A081342 A081343 * A081345 A081346 A081347

Keyword

nonn,tabl

Author

Paul Barry, Mar 19 2003

Status

approved