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 1-10 of 27 results. Next

A267111 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2*a(n-1), a(A088359(n)) = 1+2*a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 15, 10, 13, 14, 12, 16, 17, 19, 23, 31, 18, 21, 27, 22, 29, 30, 20, 25, 26, 28, 24, 32, 33, 35, 39, 47, 63, 34, 37, 43, 55, 38, 45, 59, 46, 61, 62, 36, 41, 51, 42, 53, 54, 44, 57, 58, 60, 40, 49, 50, 52, 56, 48, 64, 65, 67, 71, 79, 95, 127, 66, 69, 75, 87, 111, 70, 77, 91, 119, 78, 93, 123
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2016

Keywords

Links

Crossrefs

Inverse: A267112.
Cf. A004001, A080677, A087686, A088359, A093879, A267108, A267109.
Similar or related permutations: A006068, A054429, A276441, A233275, A233277, A276343, A276345, A276445.
Cf. also permutations A266411, A266412 and arrays A265901, A265903.

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 1 + 2*a(A004001(n)-1).
Equally, for n > 1, if A093879(n-1) = 0, a(n) = 2*a(n - A004001(n)), otherwise a(n) = 1 + 2*a(A004001(n)-1). [Above formula in a more symmetric form.]
As a composition of other permutations:
a(n) = A054429(A276441(n)).
a(n) = A233275(A276343(n)).
a(n) = A233277(A276345(n)).
a(n) = A006068(A276445(n)).
Other identities. For all n >= 0:
a(2^n) = 2^n. [Follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper.]

A267112 Permutation of natural numbers: a(1) = 1; a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 9, 12, 10, 15, 13, 14, 11, 16, 17, 21, 18, 27, 22, 24, 19, 31, 28, 29, 23, 30, 25, 26, 20, 32, 33, 38, 34, 48, 39, 42, 35, 58, 49, 51, 40, 54, 43, 45, 36, 63, 59, 60, 50, 61, 52, 53, 41, 62, 55, 56, 44, 57, 46, 47, 37, 64, 65, 71, 66, 86, 72, 76, 67, 106, 87, 90, 73, 96, 77, 80, 68, 121, 107, 109, 88
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A087686(1+n), and each right hand child as A088359(n), when their parent contains n:
|
...................1...................
2 3
4......../ \........5 7......../ \........6
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
8 9 12 10 15 13 14 11
16 17 21 18 27 22 24 19 31 28 29 23 30 25 26 20
etc.
The level k of the tree contains all numbers of binary width k, like many base-2 related permutations (A003188, A054429, etc). For a proof, see A267110, which gives the contents of each parent node (for node containing n).
A276442 shows the mirror-image of the same tree.

Links

Crossrefs

Inverse: A267111.
Cf. A000079, A000225, A004001, A006127, A087686, A088359, A267110.
Similar or related permutations: A003188, A054429, A276442, A233276, A233278, A276344, A276346, A276446.
Cf. also permutations A266411, A266412 and arrays A265901, A265903.

Formula

a(1) = 1; after which, a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(n)).
As a composition of other permutations:
a(n) = A276442(A054429(n)).
a(n) = A276344(A233276(n)).
a(n) = A276346(A233278(n)).
a(n) = A276446(A003188(n)).
Other identities. For all n >= 0:
a(2^n) = 2^n. [Follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper.]
a(A000225(n)) = A006127(n), i.e., a((2^(n+1)) - 1) = 2^n + n. [Numbers at the right edge.]

A276441 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = 1 + 2*a(n), a(A088359(n)) = 2*a(n), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 8, 13, 10, 9, 11, 31, 30, 28, 24, 16, 29, 26, 20, 25, 18, 17, 27, 22, 21, 19, 23, 63, 62, 60, 56, 48, 32, 61, 58, 52, 40, 57, 50, 36, 49, 34, 33, 59, 54, 44, 53, 42, 41, 51, 38, 37, 35, 55, 46, 45, 43, 39, 47, 127, 126, 124, 120, 112, 96, 64, 125, 122, 116, 104, 80, 121, 114, 100, 72, 113, 98
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276442.
Cf. A000079, A000225, A004001, A080677, A087686, A088359, A093879.
Related or similar permutations: A006068, A054429, A233275, A233277, A267111, A276343, A276345, A276443.
Cf. also arrays A265901, A265903.

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 1 + 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 2*a(A004001(n)-1).
As a composition of other permutations:
a(n) = A054429(A267111(n)).
a(n) = A233277(A276343(n)).
a(n) = A233275(A276345(n)).
a(n) = A006068(A276443(n)).
Other identities. For all n >= 1:
a(A000079(n-1)) = A000225(n).

A276442 Permutation of natural numbers: a(1) = 1; a(2n) = A088359(a(n)), a(2n+1) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 6, 7, 5, 4, 11, 14, 13, 15, 10, 12, 9, 8, 20, 26, 25, 30, 23, 29, 28, 31, 19, 24, 22, 27, 18, 21, 17, 16, 37, 47, 46, 57, 44, 56, 55, 62, 41, 53, 52, 61, 50, 60, 59, 63, 36, 45, 43, 54, 40, 51, 49, 58, 35, 42, 39, 48, 34, 38, 33, 32, 70, 85, 84, 105, 82, 104, 103, 120, 79, 101, 100, 119, 98, 118, 117, 126, 75, 95, 94
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A088359(n), and each right hand child as A087686(1+n), when their parent contains n:
|
...................1...................
3 2
6......../ \........7 5......../ \........4
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
11 14 13 15 10 12 9 8
20 26 25 30 23 29 28 31 19 24 22 27 18 21 17 16
etc.
As in the mirror image permutation A267112, the level k of the tree contains all numbers of binary width k like many other base-2 related permutations (A003188, A054429, A233278, etc). For a proof, see A267110, which gives the contents of each parent node (for a node containing n > 1).

Links

Crossrefs

Inverse: A276441.
Cf. A004001, A087686, A088359, A267110.
Related or similar permutations: A003188, A054429, A233276, A233278, A267112, A276344, A276346, A276444.

Programs

Formula

a(1) = 1; after which, a(2n) = A088359(a(n)), a(2n+1) = A087686(1+a(n)).
As a composition of other permutations:
a(n) = A267112(A054429(n)).
a(n) = A276344(A233278(n)).
a(n) = A276346(A233276(n)).
a(n) = A276444(A003188(n)).

A276445 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A001969(1+a(n-1)), a(A088359(n)) = A000069(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 5, 12, 13, 14, 8, 15, 11, 9, 10, 24, 25, 26, 28, 16, 27, 31, 22, 29, 19, 17, 30, 21, 23, 18, 20, 48, 49, 50, 52, 56, 32, 51, 55, 62, 44, 53, 59, 38, 57, 35, 33, 54, 61, 42, 63, 47, 45, 58, 37, 39, 34, 60, 41, 43, 46, 36, 40, 96, 97, 98, 100, 104, 112, 64, 99, 103, 110, 124, 88, 101, 107, 118, 76, 105, 115, 70
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276446.
Cf. A000069, A001969, A004001, A080677, A087686, A088359, A093879.
Similar or related permutations: A003188, A267111, A276443 (compare the scatter plots).

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A001969(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A000069(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A003188(A267111(n)).

A276343 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A005187(1+a(n)), a(A088359(n)) = A055938(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 14, 13, 12, 11, 9, 10, 8, 31, 30, 29, 28, 27, 26, 24, 20, 25, 21, 23, 22, 17, 18, 19, 16, 63, 62, 61, 60, 59, 58, 57, 55, 51, 43, 56, 52, 44, 54, 48, 53, 50, 45, 36, 47, 37, 39, 49, 40, 41, 46, 42, 33, 34, 35, 38, 32, 127, 126, 125, 124, 123, 122, 121, 120, 118, 114, 106, 90, 119, 115, 107, 91, 117, 111, 99
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276344.
Cf. A004001, A005187, A055938, A080677, A087686, A088359, A093879.
Similar or related permutations: A233276, A233278, A267111, A276345, A276441.
Compare also to the scatter-plots of A276443 and A276445.

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A005187(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A055938(a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A233276(A267111(n)).
a(n) = A233278(A276441(n)).

A276344 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A087686(1+a(n)), a(A055938(n)) = A088359(a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 13, 14, 12, 11, 10, 9, 8, 31, 28, 29, 30, 23, 25, 27, 26, 22, 24, 21, 20, 19, 18, 17, 16, 63, 59, 60, 61, 50, 52, 62, 53, 55, 56, 58, 41, 44, 49, 57, 51, 46, 54, 48, 40, 43, 47, 45, 39, 42, 38, 37, 36, 35, 34, 33, 32, 127, 122, 123, 124, 108, 110, 125, 111, 113, 114, 126, 89, 92, 117, 115, 118, 94, 119, 121
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276343.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276346, A276442.

Programs

Formula

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A087686(1+a(A213714(n)-1)), otherwise a(n) = A088359(a(A234017(n))).
As a composition of other permutations:
a(n) = A267112(A233275(n)).
a(n) = A276442(A233277(n)).

A276345 Permutation of natural numbers: a(1) = 1, a(A087686(1+n)) = A055938(a(n)), a(A088359(n)) = A005187(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 12, 10, 8, 15, 9, 11, 14, 13, 27, 23, 19, 16, 31, 21, 18, 22, 17, 26, 30, 20, 25, 24, 29, 28, 58, 53, 46, 38, 32, 63, 48, 41, 35, 42, 40, 34, 50, 33, 57, 62, 44, 39, 49, 37, 47, 45, 36, 56, 55, 61, 43, 54, 52, 51, 60, 59, 121, 113, 104, 89, 74, 64, 127, 108, 95, 81, 70, 82, 93, 79, 67, 98, 77, 66, 112, 65, 120
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276346.
Cf. A004001, A005187, A055938, A080677, A087686, A088359, A093879.
Similar or related permutations: A233276, A233278, A267111, A276343, A276441.

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A055938(a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A005187(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A233276(A276441(n)).
a(n) = A233278(A267111(n)).

A276346 Permutation of natural numbers: a(1) = 1; a(A005187(1+n)) = A088359(a(n)), a(A055938(n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 12, 9, 13, 8, 15, 14, 11, 19, 24, 22, 18, 27, 21, 23, 17, 29, 28, 25, 16, 31, 30, 26, 20, 36, 45, 43, 40, 54, 51, 35, 49, 42, 39, 41, 58, 48, 53, 34, 52, 38, 50, 44, 61, 60, 33, 59, 56, 55, 46, 32, 63, 62, 57, 47, 37, 69, 83, 81, 78, 102, 99, 74, 97, 93, 91, 68, 116, 112, 80, 88, 77, 109, 73, 75, 96, 90
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276345.
Cf. A004001, A005187, A055938, A079559, A087686, A088359, A213714, A234017.
Similar or related permutations: A233275, A233277, A267112, A276344, A276442.

Programs

Formula

a(1)=1; for n > 1, if A079559(n)=1 [when n is in A005187], a(n) = A088359(a(A213714(n)-1)), otherwise a(n) = A087686(1+a(A234017(n))).
As a composition of other permutations:
a(n) = A276442(A233275(n)).
a(n) = A267112(A233277(n)).

A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 15, 13, 14, 16, 17, 18, 20, 24, 19, 23, 30, 21, 27, 25, 22, 29, 31, 26, 28, 32, 33, 34, 36, 40, 48, 35, 39, 46, 60, 37, 43, 54, 41, 51, 49, 38, 45, 58, 47, 63, 61, 42, 53, 55, 50, 44, 57, 59, 62, 52, 56, 64, 65, 66, 68, 72, 80, 96, 67, 71, 78, 92, 120, 69, 75, 86, 108, 73, 83, 102, 81
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2016

Keywords

Links

Crossrefs

Inverse: A276444.
Cf. A000069, A001969, A004001, A080677, A087686, A088359, A093879.
Similar or related permutations: A003188, A276441, A276445 (compare the scatter plots).

Programs

Formula

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A000069(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A001969(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A003188(A276441(n)).
Showing 1-10 of 27 results. Next

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