A163254 -id:A163254 - cp's OEIS frontend

A163255 An interspersion: the order array of A163254.

1, 3, 2, 7, 5, 4, 13, 10, 8, 6, 21, 17, 14, 11, 9, 31, 26, 22, 18, 15, 12, 43, 37, 32, 27, 23, 19, 16, 57, 50, 44, 38, 33, 28, 24, 20, 73, 65, 58, 51, 45, 39, 34, 29, 25, 91, 82, 74, 66, 59, 52, 46, 40, 35, 30, 111, 101, 92, 83, 75, 67, 60, 53, 47, 41, 36

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Except for initial terms, rows 1 to 4 are A002061, A002522, A014206, A059100 and columns 1 to 4 are A002620, A024206, A014616, A004116.
This is the interspersion of the fractal sequence A167430; i.e., row n of this array consists of the numbers k such that n=A167430(k). - Clark Kimberling, Nov 03 2009

			Corner:
1....3....7...13
2....5...10...17
4....8...14...22
To obtain A163255 from A163254, replace each term of A163254 by its rank when all the terms of A163254 are arranged in increasing order.

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

Cf. A002061, A002522, A014206, A059100, A002620, A024206, A014616, A004116, A163253, A163254.

Cf. A167430. [From Clark Kimberling, Nov 03 2009]

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

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 *)

A163258 Fractal sequence of the interspersion A163257.

Original entry on oeis.org

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.

A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

Original entry on oeis.org

1, 5, 2, 11, 6, 3, 19, 12, 8, 4, 29, 20, 15, 10, 7, 41, 30, 24, 18, 14, 9, 55, 42, 35, 28, 23, 17, 13, 71, 56, 48, 40, 34, 27, 22, 16, 89, 72, 63, 54, 47, 39, 33, 26, 21, 109, 90, 80, 70, 62, 53, 46, 38, 32, 25, 131, 110, 99, 88, 79, 69, 61, 52, 45, 37, 31, 155, 132, 120, 108

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Beginning at row 6, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1.
Except for initial terms, the first seven rows are A028387, A002378, A005563, A028552, A008865, A014209, A028873, and the first column, A004652.

			Corner:
1....5...11...19
2....6...12...20
3....8...15...24
4...10...18...28
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 odd-numbered columns and then swap rows 1 and 2, leaving
3....11...23...39
4....12...24...40
6....16...30...48
10...22...38...58
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. A163253, A163254, A163255, A163256, A163258.

Let S(n,k) denote the k-th term in the n-th row. Four cases:
S(1,k)=k^2+k-1
S(2,k)=k^2+k
if n>1 is odd, then S(n,k)=k^2+(n-1)k+(n-1)(n-3)/4
if n>2 is even, then S(n,k)= k^2+(n-1)k+n(n-4)/4.

cp's OEIS Frontend

A163255 An interspersion: the order array of A163254.

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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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A163256 Fractal sequence of the interspersion A163253.

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A163258 Fractal sequence of the interspersion A163257.

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A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

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Formula