A060736 Array of square numbers read by antidiagonals in up direction.

1, 2, 4, 5, 3, 9, 10, 6, 8, 16, 17, 11, 7, 15, 25, 26, 18, 12, 14, 24, 36, 37, 27, 19, 13, 23, 35, 49, 50, 38, 28, 20, 22, 34, 48, 64, 65, 51, 39, 29, 21, 33, 47, 63, 81, 82, 66, 52, 40, 30, 32, 46, 62, 80, 100

Offset

1,2

Comments

A simple permutation of natural numbers.

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, Jan 09 2013

Links

Table of n, a(n) for n=1..55.

Boris Putievskiy, Transformations [of] 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

T(n+1, k)=n*n+k, T(k, n+1)=(n+1)*(n+1)+1-k, 1 <= k <= n+1.

a(n)=i^2-j+1 if i >= j, a(n)=(j-1)^2 + i 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). - Boris Putievskiy, Jan 09 2013

Example

1 4 9 16 .. => a(1)= 1
2 3 8 15 .. => a(2)= 2, a(3)=4
5 6 7 14 .. => a(4)= 5, a(5)=3, a(6)=9
10 11 12 13 .. => a(7)=10, a(8)=6, a(9)=8, a(10)=16

Mathematica

Table[ If[n < 2*k-1, k^2 + k - n, (n-k)^2 + k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2013 *)

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 i>=j:
   result=i**2-j+1
else:
   result=(j-1)**2+i
# Boris Putievskiy, Jan 09 2013

Crossrefs

Cf. A060734. Inverse permutation: A064788, the first inverse function (numbers of rows) A194258, the second inverse function (numbers of columns) A194195.

Sequence in context: A353708 A117606 A245814 * A097292 A269780 A038776

Adjacent sequences: A060733 A060734 A060735 * A060737 A060738 A060739

Keyword

nonn,tabl

Author

Frank Ellermann, Apr 23 2001

Status

approved