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

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, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 81, 80, 79, 78, 77

Offset

1,2

Comments

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

Contribution from Hieronymus Fischer, Apr 30 2012: (Start)

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.

A reordering of the natural numbers.

The sequence is self-inverse in that a(a(n))=n.

(End)

References

R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.

Suggested by correspondence with Michael Somos.

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..11131

Index entries for sequences that are permutations of the natural numbers

Formula

Contribution from Hieronymus Fischer, Apr 30 2012: (Start)

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.

a(n)=n for n=k(k+1)+1, k>=0.

a(n+1)=(sqrt(a(n)-1)+2)^2, if a(n)-1 is a square, a(n+1)=a(n)-1, else.

a(n)=2*(floor(sqrt(n-1))+1)*floor(sqrt(n-1))-n+2. (End)

Example

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);
a(8)=6=a(7)-1, since the minimal number not yet in the sequence (=5) is <=7=a(7).

Mathematica

Flatten[Table[Range[n^2, (n-1)^2+1, -1], {n, 10}]] (* Harvey P. Dale, Jan 10 2016 *)
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 *)

Prog

(PARI) a(n)=local(t); if(n<1, 0, t=sqrtint(n-1); 2*(t^2+t+1)-n)

Crossrefs

A self-inverse permutation of the natural numbers.

Cf. A000027, A038722.

Cf. A132666, A132664, A132665, A132674.

Sequence in context: A021235 A273991 A292828 * A084483 A266150 A276612

Adjacent sequences: A020700 A020701 A020702 * A020704 A020705 A020706

Keyword

nonn

Author

N. J. A. Sloane, May 02 2000

Status

approved