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 1-3 of 3 results.

A060170 Number of orbits of length n under the map whose periodic points are counted by A005809.

Original entry on oeis.org

3, 6, 27, 120, 600, 3078, 16611, 91872, 520749, 3004200, 17594247, 104304888, 624801957, 3775722342, 22991161500, 140928011136, 868886416866, 5384796881850, 33525472069563, 209592223788000, 1315211209630794, 8281053081282894, 52301607644921259, 331260902534858976, 2103541885645955625, 13389670112374830378
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A005809 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.
a(n) is divisible by n (cf. A268617), 2*a(n) is divisible by n^2 (cf. A268618).

Examples

			a(3) = 27 since a map whose periodic points are counted by A005809 has 3 fixed points and 84 points of period 3, hence 27 orbits of length 3.

Links

Crossrefs

Cf. A005809, A060164, A060165, A060166, A060167, A060168, A060179, A060171, A060171, A060172, A060173.

Programs

  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n; \\ Michel Marcus, Sep 10 2017

Formula

a(n) = (1/n)* Sum_{d|n} A008683(n/d)*A005809(d).

Extensions

Edited by Max Alekseyev, Feb 09 2016

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

Original entry on oeis.org

12, 3, 4, 6, 12, 25, 60, 150, 400, 1107, 3180, 9386, 28404, 87711, 275764, 880470, 2849916, 9336508, 30918732, 103384758, 348725540, 1185630123, 4060210764, 13996354586, 48541672872, 169293988125, 593488622344, 2090567755278, 7396924802052, 26281018091013, 93738717046476, 335563502259798
Offset: 1

Views

Author

Max Alekseyev, Feb 07 2016

Keywords

Links

Crossrefs

Cf. A254593, A268618, A268619.

Programs

  • Mathematica
    a[n_] := (6/n^3)* DivisorSum[n, MoebiusMu[n/#] Binomial[2 #, #] &]; Array[a, 50] (* G. C. Greubel, Dec 15 2017 *)
  • PARI
    { a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))*6/n^3; }

Formula

a(n) = A007727(n)*6/n^3 = A045630(n)*12/n^3 = A060165(n)*6/n^2 = A022553(n)*12/n^2 = A268619(n)*6/n.
For n == 0, 1, or 3 (mod 4), a(n) = 2*A254593(n); for n == 2 (mod 4), a(n) = 2*A254593(n) - A254593(n/2)/2.

A268617 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)*binomial(3*d,d).

Original entry on oeis.org

3, 3, 9, 30, 120, 513, 2373, 11484, 57861, 300420, 1599477, 8692074, 48061689, 269694453, 1532744100, 8808000696, 51110965698, 299155382325, 1764498529977, 10479611189400, 62629105220514, 376411503694677, 2273982941083533, 13802537605619124, 84141675425838225, 514987312014416553, 3163620641291970255
Offset: 1

Views

Author

Max Alekseyev, Feb 09 2016

Keywords

Comments

2*a(n) is divisible by n (cf. A268618).

Crossrefs

Cf. A005809, A008683, A060170, A268618, A268619.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[3*#, #] &] / n^2; Array[a, 30] (* Amiram Eldar, Aug 24 2023 *)
  • PARI
    { a(n) = sumdiv(n,d,moebius(n/d)*binomial(3*d,d))/n^2; }

Formula

a(n) = (1/n^2)* Sum_{d|n} A008683(n/d)*A005809(d).
a(n) = A060170(n) / n = A268618(n)*n/2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.