A090861 Permutation of natural numbers arising from a spiral.

1, 6, 5, 4, 3, 2, 9, 8, 7, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 25, 24, 23, 22, 21, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 49, 48, 47, 46, 45, 44, 43, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50

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 -> 6 -> 5 -> 4 -> 3 -> 2 -> 9 -> 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, -2, 6], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

Prog

(Sage)
def a(n):
    x = (isqrt(n-1)+1)//2
    return 8*x^2 + (-2 if n <= 4*x^2 + 2*x else 6)*x + 2 - n
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016
(PARI) {s(n)=floor((floor(sqrt(n-1)) +1)/2)};
for(n=1, 75, print1(8*s(n)^2 -n +2 +s(n)*if(n<= 2*s(n)*(2*s(n)+1), -2, 6), ", ")) \\ G. C. Greubel, Feb 05 2019

Crossrefs

Cf. A020703, A090915, A090925, A090928, A090929, A090930.

Sequence in context: A031056 A226293 A055117 * A132670 A221217 A018868

Adjacent sequences: A090858 A090859 A090860 * A090862 A090863 A090864

Keyword

easy,nonn

Author

Felix Tubiana, Feb 16 2004

Extensions

Offset corrected by Eric M. Schmidt, May 18 2016

Status

approved