cp's OEIS Frontend

A090928 Permutation of natural numbers arising from a square spiral.

%I A090928 #23 Jul 25 2025 09:00:26
%S A090928 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,
%T A090928 39,40,41,42,43,44,45,46,47,48,49,26,27,28,29,30,31,32,33,34,35,36,37,
%U A090928 66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,50,51,52,53,54,55,56
%N A090928 Permutation of natural numbers arising from a square spiral.
%C A090928 Write out the natural numbers in a square counterclockwise spiral:
%C A090928 .
%C A090928   17--16--15--14--13
%C A090928    |               |
%C A090928   18   5---4---3  12
%C A090928    |   |       |   |
%C A090928   19   6   1---2  11
%C A090928    |   |           |
%C A090928   20   7---8---9--10
%C A090928    |
%C A090928   21--22--23--24--25
%C A090928 .
%C A090928 Now read off the numbers in a counterclockwise spiral: 1 -> 6 -> 7 -> 8 -> 9 -> 2 -> 3 -> 4 -> 5 -> 18 -> etc.
%H A090928 Eric M. Schmidt, <a href="/A090928/b090928.txt">Table of n, a(n) for n = 1..1000</a>
%H A090928 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A090928 a(n) = A090925(A090925(n)). - _Rémy Sigrist_, Jul 25 2025
%t A090928 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 *)
%o A090928 (Sage)
%o A090928 def a(n):
%o A090928     x = (isqrt(n-1)+1)//2
%o A090928     return n + 4*x if n + 4*x <= (2*x+1)^2 else n - 4*x
%o A090928 [a(n) for n in (1..75)] # _Eric M. Schmidt_, May 18 2016
%o A090928 (PARI) {s(n) = ((sqrtint(n-1)+1)/2)\1};
%o A090928 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
%Y A090928 Cf. A020703, A090861, A090915, A090925, A090929, A090930.
%K A090928 easy,nonn
%O A090928 1,2
%A A090928 _Felix Tubiana_, Feb 26 2004
%E A090928 Offset corrected by _Eric M. Schmidt_, May 18 2016