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.

A166016 Inverse permutation to A138609.

Original entry on oeis.org

1, 4, 2, 3, 5, 6, 7, 8, 11, 12, 9, 10, 13, 14, 16, 17, 15, 22, 18, 19, 23, 24, 20, 21, 25, 26, 29, 30, 27, 28, 31, 32, 37, 38, 33, 34, 39, 40, 35, 36, 41, 42, 46, 47, 43, 44, 48, 49, 45, 56, 50, 51, 57, 58, 52, 53, 59, 60, 54, 55, 61, 62, 67, 68, 63, 64, 69, 70, 65, 66, 71, 72
Offset: 1

Views

Author

Antti Karttunen, Oct 05 2009

Keywords

Links

Crossrefs

Inverse: A138609.

A138612 Permutation of natural numbers generated with the sieve algorithm described in the comment lines.

Original entry on oeis.org

1, 2, 4, 3, 7, 12, 5, 11, 19, 28, 6, 15, 26, 39, 53, 8, 20, 35, 52, 71, 91, 9, 23, 42, 64, 88, 114, 141, 10, 27, 49, 76, 106, 138, 172, 207, 13, 33, 60, 93, 129, 168, 210, 253, 297, 14, 37, 68, 105, 148, 194, 243, 294, 347, 401, 16, 43, 79, 122, 171, 225, 282, 342
Offset: 1

Views

Author

Ctibor O. Zizka, May 14 2008

Keywords

Comments

Sieve proceeds as:
1) take the 1st element from natural numbers (A000027): 1; remaining set is 2,3,4,5,6,7,8,9,10,...; S1={1}
2) take the 1st element from the remaining set: 2; remaining set is 3,4,5,6,7,8,9,10,...; take the 2nd element from the remaining set: 4; remaining set is 3,5,6,7,8,9,10,...; S2={2,4}
3) take the 1st element from the remaining set: 3; remaining set is 5,6,7,8,9,10,...; take the 3rd element from the remaining set: 7; remaining set is 5,6,8,9,10,11,12,...; take the 7th element from the remaining set: 12; remaining set is 5,6,8,9,10,11,13,14,15,16,17,18,19,20,..; S3={3,7,12}
4) take the 1st element from the remaining set: 5; remaining set is 6,8,9,10,11,13,14,15,16,17,18,19,20,..; take the 5th element from the remaining set: 11; remaining set is 6,8,9,10,13,14,15,16,17,18,19,20,..; take the 11th element from the remaining set: 19; remaining set is 6,8,9,10,13,14,15,16,17,18,20,..; take the 19th element from the remaining set: 28; remaining set is 6,8,9,10,13,14,15,16,17,18,20,21,22,23,24,25,26,27,29,30,31,...;
thus S4={5,11,19,28}.
The sequence is concatenation of such subsequences S1,S2,S3,S4,S5,...,Sn, ..., where each subsequence consists of n nondecreasing terms. Alternatively, these can be viewed as rows of a triangular table.

Links

Crossrefs

Inverse: A166017. Left edge A166018, Right edge: A166019, Row sums: A166020. Cf. A138606-A138609.

Extensions

Edited, extended, keyword tabl and Scheme-code added by Antti Karttunen, Oct 05 2009

A138606 List first F(1) odd numbers, then first F(2) even numbers (starting from 2), then the next F(3) odd numbers, then the next F(4) even numbers, etc., where F(n) = A000045(n), the n-th Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 11, 13, 15, 10, 12, 14, 16, 18, 20, 22, 24, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77
Offset: 1

Views

Author

Ctibor O. Zizka, May 14 2008

Keywords

Comments

The original name was "FibCon sequence". However, this sequence has only a passing resemblance to Connell-like sequences (see A001614), which are all monotone, while this sequence is a bijection of natural numbers.
Fixed points of the permutation are the terms of A062114. - Ivan Neretin, Sep 04 2017

Examples

			Let us separate the positive integers into odd (A005408) and even numbers (A005843):
1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,...
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,...
then we get the following subsequences:
S1={1}
S2={2}
S3={3,5}
S4={4,6,8}
S5={7,9,11,13,15}
S6={10,12,14,16,18,20,22,24}
...
and concatenating them S1/S2/S3/S4/S5/... gives this sequence.

Links

Crossrefs

Inverse: A166013. A000035(a(n)) = A000035(A072649(n)). Cf. A138607-A138609, A138612.

Programs

  • Mathematica
    o = 1; e = 2; Flatten@Table[If[OddQ[n], Range[o, (o += 2 Fibonacci[n]) - 1, 2], Range[e, (e += 2 Fibonacci[n]) - 1, 2]], {n, 9}] (* Ivan Neretin, Sep 04 2017 *)

Formula

a(n) = A166012(A072649(n)-1) + 2*(n - A000045(1+A072649(n))). - Antti Karttunen, Oct 05 2009

Extensions

Edited, extended and Scheme code added by Antti Karttunen, Oct 05 2009

A138607 List first A008578(1) odd numbers, then first A008578(2) even numbers, then the next A008578(3) odd numbers, then the next A008578(4) even numbers, etc.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 14, 9, 11, 13, 15, 17, 19, 21, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73
Offset: 1

Views

Author

Ctibor O. Zizka, May 14 2008

Keywords

Comments

A permutation of numbers.

Examples

			Let
  S1={1}
  S2={2,4}
  S3={3,5,7}
  S4={6,8,10,12,14}
  S5={9,11,13,15,17,19,21}
  S6={16,18,20,22,24,26,28,30,32,34,36}
  ...
then S1, S2, S3, S4, S5, S6,... gives this sequence.

Links

Crossrefs

Inverse: A166014. Cf. A138606, A138608, A138609, A138612.

Formula

If n < 3, a(n) = n. If n-2 = A007504(A083375(n-2)), then a(n) = a(n-1-A000040(A083375(n-2)))+2, otherwise a(n) = a(n-1)+2. - Antti Karttunen, Oct 05 2009.

Extensions

Edited, extended, and offset changed from 0 to 1 by Antti Karttunen, Oct 05 2009

A138608 List first F(1) numbers from A016777, then first F(2) numbers from A016789, then the first F(3) numbers from A008585 (starting from 3), then the next F(4) numbers from A016777, then the next F(5) numbers from A016789, then the next F(6) numbers from A008585, etc, where F(n) = A000045(n), the n-th Fibonacci number.

Original entry on oeis.org

1, 2, 3, 6, 4, 7, 10, 5, 8, 11, 14, 17, 9, 12, 15, 18, 21, 24, 27, 30, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84
Offset: 1

Views

Author

Ctibor O. Zizka, May 14 2008

Keywords

Comments

The original name was "Generalized FibCon sequence". However, this sequence has only a passing resemblance to Connell-like sequences (see A001614 and the paper by Iannucci & Mills-Taylor), which are all monotone, while this sequence is a bijection of natural numbers.

Examples

			Let us separate natural numbers into three disjoint sets (A016777, A016789 and A008585):
  1,4,7,10,13,16,19,22,25,28,31,...
  2,5,8,11,14,17,20,23,26,29,32,...
  3,6,9,12,15,18,21,24,27,30,33,...
then
  S0={1}
  S1={2}
  S2={3,6}
  S3={4,7,10}
  S4={5,8,11,14,17}
  S5={9,12,15,18,21,24,27,30}
  ...
and concatenating S0/S1/S2/S3/S4/S5/... gives this sequence.

Links

Crossrefs

Inverse: A166015. A010872(a(n)) = A010872(A072649(n)). Cf. A138606-A138609, A138612.

Formula

If n < 4, a(n) = n. If n = A000045(A072649(n)+1), then a(n) = a(n-1-A000045(A072649(n)))+3, otherwise a(n) = a(n-1)+3. - Antti Karttunen, Oct 05 2009
1. The sequence is formed by concatenating subsequences S0,S1, S2, ..., each of finite length. 2. The subsequence S0 consists of the element 1. 3. The n-th subsequence has F(n) elements, F(n) denotes n-th Fibonacci number. 4. Each subsequence is nondecreasing and the difference between two consecutive elements in the same subsequence is 3.

Extensions

Edited, extended, starting offset changed from 0 to 1, and Scheme-code added by Antti Karttunen, Oct 05 2009
Showing 1-5 of 5 results.

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