cp's OEIS Frontend

A090915 Permutation of natural numbers arising from a square spiral.

%I A090915 #22 Jul 05 2022 11:12:22
%S A090915 1,8,7,6,5,4,3,2,9,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,25,48,
%T A090915 47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,49,
%U A090915 80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58
%N A090915 Permutation of natural numbers arising from a square spiral.
%C A090915 Write out the natural numbers in a square counterclockwise spiral:
%C A090915 .
%C A090915   17--16--15--14--13
%C A090915    |               |
%C A090915   18   5---4---3  12
%C A090915    |   |       |   |
%C A090915   19   6   1---2  11
%C A090915    |   |           |
%C A090915   20   7---8---9--10
%C A090915    |
%C A090915   21--22--23--24--25
%C A090915 .
%C A090915 Now read off the numbers in a square clockwise spiral: 1 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 2 -> 9 -> etc.
%H A090915 Eric M. Schmidt, <a href="/A090915/b090915.txt">Table of n, a(n) for n = 1..1000</a>
%t A090915 With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n==(2*x+1)^2, n, 8*x^2 -n+2], {n, 1, 75}]] (* _G. C. Greubel_, Feb 05 2019 *)
%o A090915 (Sage)
%o A090915 def a(n):
%o A090915     x = (isqrt(n-1)+1)//2
%o A090915     return n if n == (2*x+1)^2 else 8*x^2 + 2 - n
%o A090915 [a(n) for n in (1..75)] # _Eric M. Schmidt_, May 18 2016
%o A090915 (PARI) {s(n) = ((sqrtint(n-1)+1)/2)\1};
%o A090915 for(n=1,75, print1(if(n == (2*s(n)+1)^2, n, 8*s(n)^2-n+2), ", ")) \\ _G. C. Greubel_, Feb 05 2019
%Y A090915 Cf. A020703, A090861, A090925, A090928, A090929, A090930.
%K A090915 easy,nonn
%O A090915 1,2
%A A090915 _Felix Tubiana_, Feb 26 2004
%E A090915 Offset corrected by _Eric M. Schmidt_, May 18 2016