A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

1, 4, 2, 9, 5, 3, 16, 10, 7, 6, 25, 17, 13, 11, 8, 36, 26, 21, 18, 14, 12, 49, 37, 31, 27, 22, 19, 15, 64, 50, 43, 38, 32, 28, 23, 20, 81, 65, 57, 51, 44, 39, 33, 29, 24, 100, 82, 73, 66, 58, 52, 45, 40, 34, 30, 121, 101, 91, 83, 74, 67, 59, 53, 46, 41, 35

Offset: 1

Author

Clark Kimberling, Jul 23 2009

Keywords

• nonn
• tabl

Comments

A permutation of the natural numbers.

Row 1 consists of the squares.

Beginning with row 5, the columns obey a 3rd-order recurrence:

c(n)=c(n-1)+c(n-2)-c(n-3)+1; thus disregarding row 1, the nonsquares are partitioned by this recurrence.

Except for initial terms, the first ten rows match A000290, A002522, A002061, A059100, A014209, A117950, A027688, A087475, A027689, A117951, and the first column, A035106.

Examples

			Corner:
1....4....9...16...25
2....5...10...17...26
3....7...13...21...31
6...11...18...27...38
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the even-numbered columns, leaving
1....7...17...
2....8...18...
5...13...25...
9...19...33...
Then replace each of those numbers by its rank when all the numbers are jointly ranked.
		

Links

• Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Crossrefs

Cf. A000037, A000290, A161179, A163254, A163255, A163256, A163257, A163258.

Formula

Let S(n,k) denote the k-th term in the n-th row. Three cases:

S(1,k)=k^2;

if n is even, then S(n,k)=k^2+(n-2)k+(n^2-2*n+4)/4;

if n>=3 is odd, then S(n,k)=k^2+(n-2)k+(n^2-2*n+1)/4.

Extensions

Edited and augmented by Clark Kimberling, Jul 24 2009