A138606 -id:A138606 - cp's OEIS frontend

A166013 Inverse permutation to A138606.

1, 2, 3, 5, 4, 6, 8, 7, 9, 13, 10, 14, 11, 15, 12, 16, 21, 17, 22, 18, 23, 19, 24, 20, 25, 34, 26, 35, 27, 36, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 55, 43, 56, 44, 57, 45, 58, 46, 59, 47, 60, 48, 61, 49, 62, 50, 63, 51, 64, 52, 65, 53, 66, 54, 67, 89, 68, 90, 69, 91

Offset: 1

Antti Karttunen, Oct 05 2009

nonn

Fixed points of the permutation are the terms of A062114. - Ivan Neretin, Sep 04 2017

Inverse: A138606.

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

Ctibor O. Zizka, May 14 2008

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.

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

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

A138609 List the first term from A042963, then 2 terms from A014601 (starting from 3), 3 terms from A042963, 4 terms from A014601, etc.

Original entry on oeis.org

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

Offset: 1

Ctibor O. Zizka, May 14 2008

The original name was "Generalized Connell 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.

The sequence is formed by concatenating subsequences S1,S2,S3,..., each of finite length. The subsequence S1 consists of the element 1. The n-th subsequence has n elements. Each subsequence is nondecreasing. The difference between two consecutive elements in the same subsequence is varying, but >= 1.

			Let us separate natural numbers into two disjoint sets (A042963 and A014601):
  1,2,5,6,9,10,13,14,17,18,21,22,25,26,29,30,...
  3,4,7,8,11,12,15,16,19,20,23,24,27,28,31,32,...
then
  S1={1}
  S2={3,4}
  S3={2,5,6,}
  S4={7,8,11,12}
  S5={9,10,13,14,17}
  ...
  and concatenating S1/S2/S3/S4/S5/... gives this sequence.

Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.7
Index entries for sequences that are permutations of the natural numbers

Inverse: A166016. Cf. A138606, A138607, A138608, A138612.

a(n) = A116966(A074147(n)-1). - Antti Karttunen, Oct 05 2009

Edited, extended and keyword tabl 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

Ctibor O. Zizka, May 14 2008

A permutation of numbers.

			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.

Index entries for sequences that are permutations of the natural numbers

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

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.

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

Ctibor O. Zizka, May 14 2008

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.

			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.

Douglas E. Iannucci, Donna Mills-Taylor, On Generalizing the Connell Sequence, Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.7
Index entries for sequences that are permutations of the natural numbers

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

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.

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

A166012 a(n) = 2*(A000045(n)-(n mod 2)) + 1 + (n mod 2).

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 17, 26, 43, 68, 111, 178, 289, 466, 755, 1220, 1975, 3194, 5169, 8362, 13531, 21892, 35423, 57314, 92737, 150050, 242787, 392836, 635623, 1028458, 1664081, 2692538, 4356619, 7049156, 11405775, 18454930, 29860705, 48315634, 78176339

Offset: 0

Antti Karttunen, Oct 05 2009

This is an auxiliary sequence for computing A138606.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Table[2*Fibonacci[n] + (1 + (-1)^n)/2, {n, 0, 100}] (* G. C. Greubel, Apr 21 2016 *)
LinearRecurrence[{1,2,-1,-1},{1,2,3,4},40] (* Harvey P. Dale, May 01 2018 *)

Vec((1+x-x^2-2*x^3)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Apr 22 2016

a(2n) = 2*A000045(2n) + 1, a(2n+1) = 2*A000045(2n+1).
Without reference to A000045: a(n)=2*Floor(a(n-1)/2)+a(n-2). - Clark Kimberling, Nov 07 2009
If n mod 2 = 0 then a(n) = a(n-1) + a(n-2), else a(n) = a(n-1) + a(n-2) - 1.
a(n) = 2*Fibonacci(n) + (1+(-1)^n)/2.
a(n) = 2*Fibonacci(n) + [(n+1)mod 2]. - Gary Detlefs, Dec 29 2010
G.f.: (1 + x - x^2 - 2*x^3)/((1 - x^2)*(1 - x - x^2)). - Ilya Gutkovskiy, Apr 22 2016
From Colin Barker, Apr 22 2016: (Start)
a(n) = a(n-1)+2*a(n-2)-a(n-3)-a(n-4) for n>3.
a(n) = (1/2+(-1)^n/2-(2*((1/2*(1-sqrt(5)))^n-(1/2*(1+sqrt(5)))^n))/sqrt(5)).
(End)

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A166013 Inverse permutation to A138606.

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

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A138609 List the first term from A042963, then 2 terms from A014601 (starting from 3), 3 terms from A042963, 4 terms from A014601, etc.

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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.

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A166012 a(n) = 2*(A000045(n)-(n mod 2)) + 1 + (n mod 2).

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