A080029 -id:A080029 - cp's OEIS frontend

A079313 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

1, 3, 5, 2, 7, 8, 9, 11, 13, 12, 15, 17, 19, 16, 21, 23, 25, 20, 27, 29, 31, 24, 33, 35, 37, 28, 39, 41, 43, 32, 45, 47, 49, 36, 51, 53, 55, 40, 57, 59, 61, 44, 63, 65, 67, 48, 69, 71, 73, 52, 75, 77, 79, 56, 81, 83, 85, 60, 87, 89, 91, 64, 93, 95, 97, 68, 99, 101, 103, 72, 105

Offset: 1

J. C. Lagarias and N. J. A. Sloane, Feb 11 2003

The sequence obeys the rule: "The concatenation of a(n) and a(a(n)) is odd". Example: "1" and the 1st term, concatenated, is 11; "3" and the 3rd term, concatenated, is 35; "5" and the 5th term, concatenated, is 57; "2" and the 2nd term, concatenated, is 23; etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Equals A080032 + 1. Cf. A079000, A079250-A079259, A080029-A080031.

Rest@ CoefficientList[Series[x*(-3*x^11 + 2*x^10 - x^9 + 7*x^7 - x^6 + 2*x^5 + 5*x^4 + 2*x^3 + 5*x^2 + 3*x + 1)/(x^8 - 2*x^4 + 1), {x, 0, 120}], x] (* Michael De Vlieger, Dec 17 2024 *)

For n >= 5 a(n) is given by: a(4t-2) = 4t, a(4t-1) = 6t-3, a(4t) = 6t-1, a(4t+1) = 6t+1.
All odd numbers occur; the only even numbers which occur are 2 and the multiples of 4 excluding 4 itself.
From Chai Wah Wu, Apr 13 2024: (Start)
a(n) = 2*a(n-4) - a(n-8) for n > 12.
G.f.: x*(-3*x^11 + 2*x^10 - x^9 + 7*x^7 - x^6 + 2*x^5 + 5*x^4 + 2*x^3 + 5*x^2 + 3*x + 1)/(x^8 - 2*x^4 + 1). (End)

More terms from Matthew Vandermast, Mar 20 2003

A080032 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is even".

Original entry on oeis.org

0, 2, 4, 1, 6, 7, 8, 10, 12, 11, 14, 16, 18, 15, 20, 22, 24, 19, 26, 28, 30, 23, 32, 34, 36, 27, 38, 40, 42, 31, 44, 46, 48, 35, 50, 52, 54, 39, 56, 58, 60, 43, 62, 64, 66, 47, 68, 70, 72, 51, 74, 76, 78, 55, 80, 82, 84, 59, 86, 88, 90, 63, 92, 94, 96, 67, 98, 100, 102, 71, 104

Offset: 0

N. J. A. Sloane, Mar 14 2003

The same sequence, but without the initial 0, obeys the rule: "The concatenation of a(n) and a(a(n)) is even". Example: "2" and the 2nd term, concatenated, is 24; "4" and the 4th term, concatenated, is 46; "1" and the 1st term, concatenated, is 12; etc. - Eric Angelini, Feb 22 2017

If "even" in the definition is replaced by "prime", we get A121053. - N. J. A. Sloane, Dec 14 2024

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A079000, A080029, A080030, A121053. Equals A079313 - 1.

CoefficientList[Series[x*(-3*x^10 + 2*x^9 - x^8 + 8*x^6 + 3*x^4 + 6*x^3 + x^2 + 4*x + 2)/(x^8 - 2*x^4 + 1), {x, 0, 120}], x] (* Michael De Vlieger, Dec 17 2024 *)

For n >= 4 a(n) is given by: a(4m)=6m, a(4m+1)=4m+3, a(4m+2)=6m+2, a(4m+3)=6m+4.
From Chai Wah Wu, Apr 13 2024: (Start)
a(n) = 2*a(n-4) - a(n-8) for n > 11.
G.f.: x*(-3*x^10 + 2*x^9 - x^8 + 8*x^6 + 3*x^4 + 6*x^3 + x^2 + 4*x + 2)/(x^8 - 2*x^4 + 1). (End)

More terms from Matthew Vandermast, Mar 21 2003

A080030 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 1 mod 3".

Original entry on oeis.org

2, 1, 4, 5, 7, 10, 8, 13, 16, 11, 19, 22, 14, 25, 28, 17, 31, 34, 20, 37, 40, 23, 43, 46, 26, 49, 52, 29, 55, 58, 32, 61, 64, 35, 67, 70, 38, 73, 76, 41, 79, 82, 44, 85, 88, 47, 91, 94, 50, 97, 100, 53, 103, 106, 56, 109, 112, 59, 115, 118, 62, 121, 124, 65, 127, 130, 68

Offset: 0

N. J. A. Sloane, Mar 14 2003

A permutation of all positive non-multiples of 3. - Matthew Vandermast, Mar 21 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079313, A080029, A080031.

a(3m)=3m+2, a(3m+1)=6m+1, a(3m+2)=6m+4.

More terms from Matthew Vandermast, Mar 21 2003

A080031 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 2 mod 3".

Original entry on oeis.org

1, 2, 5, 4, 8, 11, 7, 14, 17, 10, 20, 23, 13, 26, 29, 16, 32, 35, 19, 38, 41, 22, 44, 47, 25, 50, 53, 28, 56, 59, 31, 62, 65, 34, 68, 71, 37, 74, 77, 40, 80, 83, 43, 86, 89, 46, 92, 95, 49, 98, 101, 52, 104, 107, 55, 110, 113, 58, 116, 119, 61, 122, 125, 64, 128, 131, 67

Offset: 0

N. J. A. Sloane, Mar 14 2003

A permutation of all positive non-multiples of 3; also a permutation of A080030. - Matthew Vandermast, Mar 21 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079313, A080029, A080030.

a(3m)=3m+1, a(3m+1)=6m+2, a(3m+2)=6m+5. [corrected by Georg Fischer, Jun 08 2022]

More terms from Matthew Vandermast, Mar 21 2003

cp's OEIS Frontend

A079313 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

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A080032 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is even".

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A080030 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 1 mod 3".

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A080031 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 2 mod 3".

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Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions