A090930 Permutation of natural numbers arising from a square spiral.

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

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 clockwise spiral: 1 -> 2 -> 9 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 12 -> 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[8*x^2 -n +2 +x*If[n <= 4*x^2 -2*x, -6, 2], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

Prog

(Sage)
def a(n):
    x = (isqrt(n-1)+1)//2
    return 8*x^2 + (-6 if n <= 4*x^2 - 2*x else 2)*x + 2 - n
[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(8*s(n)^2 -n +2 +s(n)*if(n <= 2*s(n)*(2*s(n)-1), -6, 2), ", ")) \\ G. C. Greubel, Feb 05 2019

Crossrefs

Cf. A020703, A090861, A090915, A090925, A090928, A090929.

Sequence in context: A018799 A076219 A077601 * A346429 A258412 A151927

Adjacent sequences: A090927 A090928 A090929 * A090931 A090932 A090933

Keyword

easy,nonn

Author

Felix Tubiana, Feb 26 2004

Extensions

Offset corrected by Eric M. Schmidt, May 18 2016

Status

approved