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A020703 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,3,5,7,...

%I A020703 #24 Jan 28 2019 19:19:45
%S A020703 1,4,3,2,9,8,7,6,5,16,15,14,13,12,11,10,25,24,23,22,21,20,19,18,17,36,
%T A020703 35,34,33,32,31,30,29,28,27,26,49,48,47,46,45,44,43,42,41,40,39,38,37,
%U A020703 64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,81,80,79,78,77
%N A020703 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,3,5,7,...
%C A020703 Arrange A000027, the natural numbers, into a (square) spiral, say clockwise as shown in A068225. Read the numbers from the resulting counterclockwise spiral of the same shape that also begins with 1 and this sequence results. - _Rick L. Shepherd_, Aug 04 2006
%C A020703 Contribution from _Hieronymus Fischer_, Apr 30 2012: (Start)
%C A020703 The sequence may also be defined as follows: a(1)=1, a(n)=m^2 (where m^2 is the least square > a(k) for 1<=k<n), if the minimal natural number not yet in the sequence is greater than a(n-1), else a(n)=a(n-1)-1.
%C A020703 A reordering of the natural numbers.
%C A020703 The sequence is self-inverse in that a(a(n))=n.
%C A020703 (End)
%D A020703 R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.
%D A020703 Suggested by correspondence with Michael Somos.
%H A020703 Hieronymus Fischer, <a href="/A020703/b020703.txt">Table of n, a(n) for n = 1..11131</a>
%H A020703 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A020703 Contribution from _Hieronymus Fischer_, Apr 30 2012: (Start)
%F A020703 a(n)=a(n-1)-1, if a(n-1)-1 > 0 is not in the set {a(k)| 1<=k<n}, else a(n)=m^2, where m^2 is the least square not yet in the sequence.
%F A020703 a(n)=n for n=k(k+1)+1, k>=0.
%F A020703 a(n+1)=(sqrt(a(n)-1)+2)^2, if a(n)-1 is a square, a(n+1)=a(n)-1, else.
%F A020703 a(n)=2*(floor(sqrt(n-1))+1)*floor(sqrt(n-1))-n+2. (End)
%e A020703 a(2)=4=2^2, since 2^2 is the least square >2=a(1) and the minimal number not yet in the sequence is 2>1=a(1);
%e A020703 a(8)=6=a(7)-1, since the minimal number not yet in the sequence (=5) is <=7=a(7).
%t A020703 Flatten[Table[Range[n^2,(n-1)^2+1,-1],{n,10}]] (* _Harvey P. Dale_, Jan 10 2016 *)
%t A020703 With[{nn=20},Flatten[Reverse/@TakeList[Range[nn^2],Range[1,nn,2]]]] (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Jan 28 2019 *)
%o A020703 (PARI) a(n)=local(t); if(n<1,0,t=sqrtint(n-1); 2*(t^2+t+1)-n)
%Y A020703 A self-inverse permutation of the natural numbers.
%Y A020703 Cf. A000027, A038722.
%Y A020703 Cf. A132666, A132664, A132665, A132674.
%K A020703 nonn
%O A020703 1,2
%A A020703 _N. J. A. Sloane_, May 02 2000