A093210 -id:A093210 - cp's OEIS frontend

A092964 Numbers > 1 in A051168, with a(0) = 1.

1, 2, 2, 2, 3, 2, 3, 5, 5, 3, 3, 7, 8, 7, 3, 4, 9, 14, 14, 9, 4, 4, 12, 20, 25, 20, 12, 4, 5, 15, 30, 42, 42, 30, 15, 5, 5, 18, 40, 66, 75, 66, 40, 18, 5, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 6, 26, 70, 143, 212, 245, 212, 143, 70, 26, 6, 7, 30, 91, 200, 333, 429, 429, 333, 200

Offset: 0

Thomas O. Hoffbauer (Thomas.Hoffbauer(AT)cibamberg.de), Apr 20 2004

Original definition: Triangle read by rows in which row n gives the number of equal configurations under cyclic shift.
T(n,k) = A051168(n+3,k+1), if 0Michael Somos, Jul 17 2004
From Paul Weisenhorn, Dec 21 2010 (Start):
T(n,k) is the number of classes that have (k+1) ordered sums of (n+4) with (k+1) positive integers, that can be transformed into each other by a cyclic permutation.
m has 2^(m-1) ordered sums; for each sum one remove the first part z(1) and add 1 to the next z(1) parts to get a new ordered sum until a period is reached. T(n,k)=a(m) with m=(n+4)*(n+3)/2+k+1 gives for m the number of periods with length (n+4).
The numbers m=n*(n+3)/2 with 1<=n have one period with length (n+1).
The numbers m=n*(n+3)/2+2 with 1<=n have one period with length (n+2).
The triangular numbers n*(n+1)/2 with 1<=n have one period [(n+(n-1)+...+2+1)] with length 1. (End)

			As triangle, starts:
  1;
  2,2;
  2,3,2;
  3,5,5,3;
  3,7,8,7,3;
  4,9,14,14,9,4;
  4,12,20,25,20,12,4; ...
From _Paul Weisenhorn_, Dec 21 2010: (Start)
T(2,2)=3 classes with 3 ordered sums of 6; [(1+1+4),(1+4+1),(4+1+1)]; [(1+2+3),(2+3+1),(3+1+2)]; [(1+3+2),(3+2+1),(2+1+3)].
T(2,2)=a(m)=3 periods with length 6 for m=6*5/2+3=18 [(5+5+4+3+1),(6+5+4+2+1),(6+5+3+2+1+1),(6+4+3+2+2+1),(5+4+3+3+2+1),(5+4+4+3+2)]; [(5+5+3+3+2),(6+4+4+3+1),(5+5+4+2+1+1),(6+5+3+2+2),(6+4+3+3+1+1),(5+4+4+2+2+1)]; [(5+5+3+2+2+1),(6+4+3+3+2),(5+4+4+3+1+1),(5+5+4+2+2),(6+5+3+3+1),(6+4+4+2+1+1)]. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

Row sums give A093210. Essentially the same as A051168. See A185158 for another version.

T[n_, k_] := DivisorSum[GCD[k + 1, n + 4], Binomial[(n + 4)/#, (k + 1)/#] * MoebiusMu[#] & ]/(n + 4); Table[T[n, k], {n, 0, 12}, {k, 1, n + 1}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)

T(n,k)=local(A,ps,c); n+=3; k++; if(k<1||k>=n-1, 0, A=x*O(x^n) + y*O(y^n); ps=1-x-y+A; for(m=1,n,for(i=0,m,c=polcoeff(polcoeff(ps,i,x),m-i, y); if(m==n&i==k,break(2),ps*=(1-y^(m-i)*x^i+A)^c)));-c) /* Michael Somos, Jul 17 2004 */

Edited with better definition by Omar E. Pol, Jan 05 2009

A060477 Number of orbits of length n in map whose periodic points are A000051.

Original entry on oeis.org

3, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008, 9586395, 18512790, 35790267, 69273666, 134215680, 260300986, 505286415, 981706806, 1908866960, 3714566310

Offset: 1

Thomas Ward

			a(3)=2 since the 3rd term of A000051 is 9 and the first term is 3.

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. A000051.
Cf. A001037, A059966 (both nearly identical to this sequence).
Cf. A093210.

a000051(n) = 2^n+1;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a000051(n/d)); \\ Michel Marcus, Sep 11 2017

from sympy import mobius, divisors
def A060477(n): return sum(mobius(n//d)*(2**d+1) for d in divisors(n,generator=True))//n # Chai Wah Wu, Feb 03 2022

a(n) = (1/n)* Sum_{d|n} mu(d)*A000051(n/d).

A048578 replaced by A000051 in name and formula by Michel Marcus, Sep 11 2017

cp's OEIS Frontend

A092964 Numbers > 1 in A051168, with a(0) = 1.

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