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 1-4 of 4 results.

A268712 Permutation of natural numbers: a(1) = 1, a(2n) = A268677(a(n)), a(2n+1) = A268678(1+a(n)).

Original entry on oeis.org

1, 2, 3, 6, 4, 9, 5, 13, 11, 10, 7, 21, 18, 12, 8, 29, 23, 25, 20, 24, 19, 14, 15, 45, 40, 39, 34, 28, 22, 17, 16, 62, 56, 49, 42, 54, 48, 44, 37, 51, 47, 43, 36, 30, 26, 33, 27, 95, 87, 84, 75, 80, 74, 73, 65, 61, 53, 46, 41, 38, 32, 35, 31, 129, 120, 115, 108, 100, 93, 88, 82, 112, 105, 99, 92, 94, 86, 78, 70
Offset: 1

Views

Author

Antti Karttunen, Feb 11 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A268677(n), and each right hand child as A268678(1+n), when the parent node contains n:
|
...................1...................
2 3
6......../ \........4 9......../ \........5
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
13 11 10 7 21 18 12 8
29 23 25 20 24 19 14 15 45 40 39 34 28 22 17 16
etc.

Links

Crossrefs

Inverse: A268711.
Cf. A268677, A268678.

Formula

a(1) = 1, after which: a(2n) = A268677(a(n)), a(2n+1) = A268678(1+a(n)).

A268711 Permutation of natural numbers: a(1) = 1, a(A268677(n)) = 2*a(n), a(A268678(n+1)) = 1+(2*a(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 11, 15, 6, 10, 9, 14, 8, 22, 23, 31, 30, 13, 21, 19, 12, 29, 17, 20, 18, 45, 47, 28, 16, 44, 63, 61, 46, 27, 62, 43, 39, 60, 26, 25, 59, 35, 42, 38, 24, 58, 41, 37, 34, 91, 40, 95, 57, 36, 90, 33, 89, 94, 127, 123, 56, 32, 88, 93, 55, 125, 126, 122, 87, 79, 92, 121, 54, 53, 51, 124, 86, 78, 120
Offset: 1

Views

Author

Antti Karttunen, Feb 11 2016

Keywords

Links

Crossrefs

Inverse: A268712.
Cf. A268677, A268678, A268680.

Formula

a(1) = 1, for n > 1: if A268680(n) = A268680(n-1) [when n is in A268677] a(n) = 2*a(n-A268680(n)), otherwise [when n is in A268678] a(n) = 1 + 2*a(A268680(n)-1).

A268395 Partial sums of A268389.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 4, 4, 5, 7, 7, 8, 8, 8, 11, 11, 15, 16, 16, 18, 18, 18, 19, 20, 20, 20, 22, 22, 23, 26, 26, 26, 27, 31, 31, 32, 32, 32, 34, 36, 36, 36, 37, 37, 40, 41, 41, 42, 42, 42, 47, 47, 48, 50, 50, 50, 52, 53, 53, 56, 56, 56, 57, 57, 59, 60, 60, 64, 64, 64, 65, 66, 66, 66, 69, 69, 70, 72, 72, 74, 74, 74, 75, 75, 81
Offset: 0

Views

Author

Antti Karttunen, Feb 10 2016

Keywords

Links

Crossrefs

Cf. A048631, A268389, A268672.
Cf. A268678 (with duplicates removed), A268677 (numbers that do not occur here).
Cf. A268708, A268709, A268710.
Cf. also A054861.

Programs

  • Mathematica
    f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Accumulate@ Array[f, {85}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

Formula

a(0) = 0, for n >= 1, a(n) = A268389(n) + a(n-1).
Other identities. For all n >= 0:
a(n) = A268389(A048631(n)).
a(n) = n - A268672(n).

A268678 Distinct values in A268395; partial sums of A268679.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 11, 15, 16, 18, 19, 20, 22, 23, 26, 27, 31, 32, 34, 36, 37, 40, 41, 42, 47, 48, 50, 52, 53, 56, 57, 59, 60, 64, 65, 66, 69, 70, 72, 74, 75, 81, 82, 83, 86, 87, 89, 90, 92, 93, 98, 101, 102, 104, 105, 106, 108, 109, 113, 116, 117, 119, 120, 121, 123, 124, 127, 131, 132, 134, 135, 136, 138, 139, 142
Offset: 0

Views

Author

Antti Karttunen, Feb 10 2016

Keywords

Links

Crossrefs

Cf. A001969, A268395, A268679.
Cf. A268677 (complement).
Cf. A268680 (least monotonic left inverse).
Cf. A268712.
Cf. also A004128.

Programs

  • Mathematica
    f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Union@ Accumulate@ Array[f, {150}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

Formula

a(0) = 0, for n >= 1, a(n) = A268679(n) + a(n-1).
a(n) = A268395(A001969(1+n)).
Other identities. For all n >= 0:
A268680(a(n)) = n.
Showing 1-4 of 4 results.

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