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 1-8 of 8 results.

A060165 Number of orbits of length n under the map whose periodic points are counted by A000984.

Original entry on oeis.org

2, 2, 6, 16, 50, 150, 490, 1600, 5400, 18450, 64130, 225264, 800046, 2865226, 10341150, 37566720, 137270954, 504171432, 1860277042, 6892317200, 25631327190, 95640829922, 357975249026, 1343650040256, 5056424257500, 19073789328750, 72108867614796
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A000984 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.
The number of n-cycles in the graph of overlapping m-permutations where n <= m. - Richard Ehrenborg, Dec 10 2013
a(n) is divisible by n (cf. A268619), 6*a(n) is divisible by n^2 (cf. A268592). - Max Alekseyev, Feb 09 2016
Apparently the number of Lyndon words of length n with a 4-letter alphabet (see A027377) where the first letter of the alphabet appears with the same frequency as the second of the alphabet. E.g a(1)=2 counts the words (2), (3), a(2)= 2 counts (01) (23), a(3)=6 counts (021) (031) (012) (013) (223) (233). R. J. Mathar, Nov 04 2021

Examples

			a(5) = 50 because if a map has A000984 as its periodic points, then it would have 2 fixed points and 252 points of period 5, hence 50 orbits of length 5.

Links

Crossrefs

Cf. A000984, A007727, A060164, A060166, A060167, A060168, A060169, A060170, A060171, A060172, A060173.

Programs

  • Maple
    with(numtheory):
a:= n-> add(mobius(n/d)*binomial(2*d, d), d=divisors(n))/n:
seq(a(n), n=1..30); # Alois P. Heinz, Dec 10 2013
  • Mathematica
    a[n_] := (1/n)*Sum[MoebiusMu[d]*Binomial[2*n/d, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jul 16 2015 *)
  • PARI
    a(n)=sumdiv(n,d,moebius(n/d)*binomial(2*d,d))/n \\ Charles R Greathouse IV, Dec 10 2013
  • Python
    from sympy import mobius, binomial, divisors
def a(n): return sum(mobius(n//d) * binomial(2*d, d) for d in divisors(n))//n
print([a(n) for n in range(1, 31)])  # Indranil Ghosh, Jul 24 2017

Formula

a(n) = (1/n) * Sum_{d|n} mu(d) A000984(n/d) with mu = A008683.
a(n) = 2*A022553(n).
a(n) = A007727(n)/n. - R. J. Mathar, Jul 24 2017
G.f.: 2 * Sum_{k>=1} mu(k)*log((1 - sqrt(1 - 4*x^k))/(2*x^k))/k. - Ilya Gutkovskiy, May 18 2019
a(n) ~ 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 04 2022

A060171 Number of orbits of length n under a map whose periodic points seem to be counted by A006953.

Original entry on oeis.org

12, 54, 80, 30, 24, 5400, 0, 990, 1568, 636, 24, 2720, 0, 240, 5704, 510, 0, 3835776, 0, 26724, 3600, 108, 24, 89760, 0, 240, 1064, 120, 24, 113569300, 0, 510, 11752, 0, 264, 278281640
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A006953 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.

Examples

			u(3) = 80 since a map whose periodic points are counted by A006953 has 12 fixed points and 252 points of period 3, hence 80 orbits of length 3.

Links

Crossrefs

Cf. A006953, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060172, A060173.

Programs

  • PARI
    a(n) = sumdiv(n, d, moebius(d)*denominator(bernfrac(2*n/d)/(2*n/d)))/n; \\ Michel Marcus, Sep 10 2017

Formula

a(n) = (1/n)* Sum_{d|n} mu(d)*A006953(n/d).

A060164 Number of orbits of length n under the map whose periodic points are counted by A000364.

Original entry on oeis.org

1, 2, 20, 345, 10104, 450450, 28480140, 2423938845, 267208852820, 37037118818700, 6304443126648900, 1292877846962865230, 314390193022547991720, 89447117243116404721950, 29436259549934873636908816, 11094961973721205588579579845, 4748429366816935180127543967840
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A000364 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.

Examples

			u(3) = 20 since the conjectured map whose periodic points are counted by A000364 would have 1 fixed point and 61 points of period 3, so it must have 20 orbits of length 3.

Links

Crossrefs

Cf. A000364, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060172, A060173.

Formula

a(n) = (1/n)* Sum_{d|n} mu(d)*A000364(n/d).

A060166 Number of orbits of length n under the map whose periodic points are counted by A001641.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 10, 17, 26, 44, 68, 115, 184, 306, 500, 835, 1374, 2301, 3822, 6409, 10718, 18028, 30280, 51077, 86130, 145641, 246370, 417600, 708246, 1203069, 2045010, 3480408, 5927660, 10105819, 17241140, 29439580, 50302162, 86012630, 147166248, 251963055, 431633348
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A001641 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.

Examples

			a(7) = 7 since a map whose periodic points are counted by A001641 would have 1 fixed point and 50 points of period 7, hence 7 orbits of length 7.

Links

Crossrefs

Cf. A001641, A060164, A060165, A060167, A060168, A060169, A060170, A060171, A060171.

Programs

  • PARI
    a001641(n)=if(n<0, 0, polcoeff(x*(1+2*x+4*x^3)/(1-x-x^2-x^4)+x*O(x^n), n))
a(n) = sumdiv(n, d, moebius(d)*a001641(n/d))/n; \\ Michel Marcus, Sep 10 2017

Formula

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001641(n/d).

Extensions

More terms from Michel Marcus, Sep 10 2017

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

Original entry on oeis.org

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

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Extensions

More terms from Michel Marcus, Sep 11 2017

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

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Extensions

More terms from Michel Marcus, Sep 11 2017

A060172 Number of orbits of length n under a map whose periodic points are counted by A027306.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 19, 28, 62, 93, 205, 315, 703, 1091, 2440, 3855, 8616, 13797, 30801, 49929, 111311, 182361, 405751, 671088, 1490409, 2485504, 5509504, 9256395, 20480421, 34636833, 76499520, 130150493, 286960946, 490853403, 1080476338, 1857283155, 4081876927, 7048151355
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A027306 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.

Examples

			u(7) = 9 since the map whose periodic points are counted by A027306 has 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.

Links

Crossrefs

Cf. A027306, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060173.

Programs

  • PARI
    a027306(n) = (2^n + if(n%2, 0, binomial(n, n/2)))/2;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a027306(n/d)); \\ Michel Marcus, Sep 11 2017

Formula

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A027306(n/d).

Extensions

More terms from Michel Marcus, Sep 11 2017

A060173 Number of orbits of length n under a map whose periodic points are counted by A056045.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 12, 10, 30, 1, 139, 1, 252, 231, 920, 1, 3780, 1, 10250, 5601, 32076, 1, 149390, 2126, 400036, 173692, 1475642, 1, 6196651, 1, 19113136, 5864915, 68635494, 201405, 289525026, 1, 930138540, 208267554, 3469290971, 1, 14075005210, 1, 47994721225, 7683440470
Offset: 1

Views

Author

Thomas Ward, Mar 13 2001

Keywords

Comments

The sequence A056045 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.

Examples

			a(7) = 1 since the map whose periodic points are counted by A056045 has 1 fixed point and 8 points of period 7, hence 1 orbits of length 7.

Links

Crossrefs

Cf. A056045, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060172.

Programs

  • PARI
    a056045(n) = sumdiv(n, d, binomial(n, d));
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a056045(n/d)); \\ Michel Marcus, Sep 11 2017

Formula

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A056045(n/d).

Extensions

More terms from Michel Marcus, Sep 11 2017
Showing 1-8 of 8 results.

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

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