A163257 -id:A163257 - cp's OEIS frontend

A163258 Fractal sequence of the interspersion A163257.

1, 2, 3, 4, 1, 2, 5, 3, 6, 4, 1, 2, 7, 5, 3, 8, 6, 4, 1, 2, 9, 7, 5, 3, 10, 8, 6, 4, 1, 2, 11, 9, 7, 5, 3, 12, 10, 8, 6, 4, 1, 2, 13, 11, 9, 7, 5, 3, 14, 12, 10, 8, 6, 4, 1, 2, 15, 13, 11, 9, 7, 5, 3, 16, 14, 12, 10, 8, 6, 4, 1, 2, 17, 15, 13, 11, 9, 7, 5, 3, 18, 16, 14, 12, 10, 8, 6, 4

Offset: 1

Clark Kimberling, Jul 24 2009

nonn

As a fractal sequence, A163258 contains every positive integer; indeed, A163258 properly contains itself (infinitely many times).

			Append the following segments:
1 2 3 4
1 2 5 3 6 4
1 2 7 5 3 8 6 4
1 2 9 7 5 3 10 8 6 4
For n>1, the n-th segment arises from the (n-1)st by inserting 2*n+1 at position 3 and 2*n+2 at position n+3.

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

Cf. A163253, A163254, A163255, A163256, A163258.

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

Original entry on oeis.org

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

Clark Kimberling, Jul 23 2009

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.

			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.

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

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

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.

Edited and augmented by Clark Kimberling, Jul 24 2009

A163254 Array of the nonsquares; the columns satisfy c(n)=c(n-1)+c(n-2)-c(n-3)+1.

Original entry on oeis.org

2, 5, 3, 10, 7, 6, 17, 13, 11, 8, 26, 21, 18, 14, 12, 37, 31, 27, 22, 19, 15, 50, 43, 38, 32, 28, 23, 20, 65, 57, 51, 44, 39, 33, 29, 24, 82, 73, 66, 58, 52, 45, 40, 34, 30, 101, 91, 83, 74, 67, 59, 53, 46, 41, 35, 122, 111, 102, 92, 84, 75, 68, 60, 54

Offset: 1

Clark Kimberling, Jul 23 2009

This is the array remaining after row 1 is expelled from the array in A163253.

			Corner:
2....5...10...17...26
3....7...13...21...31
6...11...18...27...38

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

Cf. A000290, A163253, A163255, A163256, A163257, A163258.

Let S(n,k) for the k-th term in the n-th row.
if n is odd, then S(n,k)=k^2+(n-1)k+(n^2+3)/4;
if n is even, then S(n,k)=k^2+(n-1)k+(n^2)/4.

A163256 Fractal sequence of the interspersion A163253.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 3, 5, 1, 2, 4, 6, 3, 5, 7, 1, 2, 4, 6, 8, 3, 5, 7, 9, 1, 2, 4, 6, 8, 10, 3, 5, 7, 9, 11, 1, 2, 4, 6, 8, 10, 12, 3, 5, 7, 9, 11, 13, 1, 2, 4, 6, 8, 10, 12, 14, 3, 5, 7, 9, 11, 13, 15, 1, 2, 4, 6, 8, 10, 12, 14, 16, 3, 5, 7, 9, 11, 13, 15, 17, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 3, 5, 7

Offset: 1

Clark Kimberling, Jul 24 2009

nonn

As a fractal sequence, A163256 contains every positive integer; indeed, A163256 properly contains itself (infinitely many times).

			Append the following segments:
  1 2 3
  1 2 4 3 5
  1 2 4 6 3 5 7
  1 2 4 6 8 3 5 7 9
For n>1, the n-th segment arises from the (n-1)st by inserting 2*n at position n+1 and appending 2*n+1 at position 2*n+1.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A163253, A163254, A163255, A163257, A163258.

Flatten[FoldList[Append[Insert[#1, 2 #2, #2 + 1], 2 #2 + 1] &, {1}, Range[10]]] (* Birkas Gyorgy, Jul 09 2012 *)

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A163258 Fractal sequence of the interspersion A163257.

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

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A163254 Array of the nonsquares; the columns satisfy c(n)=c(n-1)+c(n-2)-c(n-3)+1.

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A163256 Fractal sequence of the interspersion A163253.

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