A090928 Permutation of natural numbers arising from a square spiral.

1, 6, 7, 8, 9, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 24, 25, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 50, 51, 52, 53, 54, 55, 56

Offset

1,2

Comments

Write out the natural numbers in a square counterclockwise spiral:

.

17--16--15--14--13

| |

18 5---4---3 12

| | | |

19 6 1---2 11

| | |

20 7---8---9--10

|

21--22--23--24--25

.

Now read off the numbers in a counterclockwise spiral: 1 -> 6 -> 7 -> 8 -> 9 -> 2 -> 3 -> 4 -> 5 -> 18 -> etc.

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Mathematica

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n +4*x <= (2*x+1)^2, n+4*x, n-4*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

Prog

(Sage)
def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 4*x if n + 4*x <= (2*x+1)^2 else n - 4*x
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016
(PARI) {s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1, 75, print1(if(n+4*s(n) <= (2*s(n)+1)^2, n +4*s(n), n - 4*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

Crossrefs

Cf. A020703, A090861, A090915, A090925, A090929, A090930.

Sequence in context: A202703 A308147 A308165 * A199174 A335094 A153604

Adjacent sequences: A090925 A090926 A090927 * A090929 A090930 A090931

Keyword

easy,nonn

Author

Felix Tubiana, Feb 26 2004

Extensions

Offset corrected by Eric M. Schmidt, May 18 2016

Status

approved