A001642 -id:A001642 - cp's OEIS frontend

A060167 Number of orbits of length n under the map whose periodic points are counted by A001642.

1, 1, 1, 2, 4, 5, 9, 13, 23, 36, 63, 101, 175, 290, 497, 840, 1445, 2460, 4247, 7293, 12619, 21805, 37856, 65695, 114401, 199280, 347944, 607959, 1064130, 1864083, 3269948, 5740840, 10090148, 17748870, 31250297, 55063603, 97102485, 171355485, 302605780, 534729160, 945513850

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001642 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(7) = 9 since a map whose periodic points are counted by A001642 would have 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001642, A060164, A060165, A060166, A060168, A060169, A060170, A060171, A060171.

a001642(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4)/(1-x-x^2-x^4-x^5)+x*O(x^n), n));
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001642(n/d)); \\ Michel Marcus, Sep 11 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001642(n/d).

More terms from Michel Marcus, Sep 11 2017

A113676 Number of elements of rows of Golomb's sequence A001462, with one less 2, interpreted as triangle: Start with first row 1. The row sum of row n-1 gives the number of elements taken from A001642 (one less 2) of row n.

Original entry on oeis.org

1, 1, 2, 6, 27, 234, 6202, 1084009, 4362192095

Offset: 1

Floor van Lamoen and Paul D. Hanna, Nov 06 2005

a(n+1) gives row sum of row n of this triangle.
Conjecture: a(n) for n>1 gives first differences of Lionel Levine's sequence A014644(n) for n>=3.
Conjecture: Final elements of the rows form A014644 except for duplicate 2.

			The triangle begins
  1;
  2;
  3,3;
  4,4,4,5,5,5;
  ...
Row 4: [4,4,4,5,5,5] is generated from row 3: [3,3] because there are (3) 4's and (3) 5's in row 4.

A060168 Number of orbits of length n under the map whose periodic points are counted by A001643.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 10, 15, 26, 42, 74, 121, 212, 357, 620, 1064, 1856, 3209, 5618, 9794, 17192, 30153, 53114, 93554, 165308, 292250, 517802, 918207, 1630932, 2899434, 5161442, 9196168, 16402764, 29281168, 52319364, 93555601, 167427844, 299841117, 537357892, 963641588, 1729192432

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001643 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(7) = 10 since a map whose periodic points are counted by A001643 would have 1 fixed point and 71 points of period 7, hence 10 orbits of length 7.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001642, A060164, A060165, A060166, A060167, A060169, A060170, A060171, A060171.

a001643(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4+6*x^5)/(1-x-x^2-x^4-x^5-x^6)+x*O(x^n), n))
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001643(n/d)); \\ Michel Marcus, Sep 11 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001643(n/d).

More terms from Michel Marcus, Sep 11 2017

A060169 Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 4, 4, 5, 8, 6, 12, 13, 16, 23, 26, 35, 46, 54, 76, 89, 120, 154, 192, 255, 322, 411, 544, 679, 898, 1145, 1476, 1925, 2466, 3201, 4156, 5338, 6978, 8985

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001945 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(17) = 8 since the map whose periodic points are counted by A001945 has 1 fixed point and 137 points of period 17, hence 8 orbits of length 7.

Manfred Einsiedler, Graham Everest and Thomas Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001642, A060164, A060165, A060166, A060167, A060168, A060170, A060171, A060171, A060172, A060173.

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001945(n/d).

cp's OEIS Frontend

A060167 Number of orbits of length n under the map whose periodic points are counted by A001642.

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A113676 Number of elements of rows of Golomb's sequence A001462, with one less 2, interpreted as triangle: Start with first row 1. The row sum of row n-1 gives the number of elements taken from A001642 (one less 2) of row n.

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A060168 Number of orbits of length n under the map whose periodic points are counted by A001643.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A060169 Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula