cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Clark Kimberling, Jul 24 2009

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

A163255 An interspersion: the order array of A163254.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 24 2009

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A002061, A002522, A014206, A059100, A002620, A024206, A014616, A004116, A163253, A163254.
Cf. A167430. [From Clark Kimberling, Nov 03 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

Views

Author

Clark Kimberling, Jul 23 2009

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Formula

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.

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

Views

Author

Clark Kimberling, Jul 24 2009

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Views

Author

Clark Kimberling, Jul 24 2009

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Formula

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.
Showing 1-5 of 5 results.

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