|
| |
|
|
A000358
|
|
Number of binary necklaces of length n with no subsequence 00, excluding the necklace "0".
|
|
22
|
|
|
|
1, 2, 2, 3, 3, 5, 5, 8, 10, 15, 19, 31, 41, 64, 94, 143, 211, 329, 493, 766, 1170, 1811, 2787, 4341, 6713, 10462, 16274, 25415, 39651, 62075, 97109, 152288, 238838, 375167, 589527, 927555, 1459961, 2300348, 3626242, 5721045, 9030451, 14264309, 22542397, 35646312, 56393862, 89264835, 141358275
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
a(n) is also the number of inequivalent compositions of n into parts 1 and 2 where two compositions are considered to be equivalent if one is a cyclic rotation of the other. a(6)=5 because we have: 2+2+2, 2+2+1+1, 2+1+2+1, 2+1+1+1+1, 1+1+1+1+1+1. - Geoffrey Critzer, Feb 01 2014
|
|
|
REFERENCES
|
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 499.
T. Helleseth and A. Kholosha, Bent functions and their connections to combinatorics, in Surveys in Combinatorics 2013, edited by Simon R. Blackburn, Stefanie Gerke, Mark Wildon, Camb. Univ. Press, 2013.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = (1/n) * Sum_{ d divides n } totient(n/d) [ Fib(d-1)+Fib(d+1) ].
G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x)=x*(1+x). - Joerg Arndt, Aug 06 2012
a(n) = Sum_{0 <= i <= ceiling((n-1)/2)} [ (1/(n - i)) * Sum_{d|gcd(i, n-i)} phi(d) * binomial((n - i)/d, i/d) ]. (This is DeFord's formula for the number of distinct Lucas tilings of a 1 X n bracelet up to symmetry, even though in the paper he refers to sequence A032192(n) = a(n) - 1.) - Petros Hadjicostas, Jun 07 2019
|
|
|
EXAMPLE
|
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 5*x^7 + 8*x^8 + 10*x^9 + ... - Michael Somos, Jun 02 2019
Binary necklaces are: 1; 01, 11; 011, 111; 0101, 0111, 1111; 01010, 01011, 01111. - Michael Somos, Jun 02 2019
|
|
|
MAPLE
|
A000358 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + phi(n/d)*(fibonacci(d+1)+fibonacci(d-1)) od; RETURN(sum/n); end;
with(combstruct); spec := {A=Union(zero, Cycle(one), Cycle(Prod(zero, Sequence(one, card>0)))), one=Atom, zero=Atom}; seq(count([A, spec, unlabeled], size=i), i=1..30);
|
|
|
MATHEMATICA
|
nn=48; Drop[Map[Total, Transpose[Map[PadRight[#, nn]&, Table[ CoefficientList[ Series[CycleIndex[CyclicGroup[n], s]/.Table[s[i]->x^i+x^(2i), {i, 1, n}], {x, 0, nn}], x], {n, 0, nn}]]]], 1] (* Geoffrey Critzer, Feb 01 2014 *)
max = 50; B[x_] := x*(1+x); A = Sum[EulerPhi[k]/k*Log[1/(1-B[x^k])], {k, 1, max}]/x + O[x]^max; CoefficientList[A, x] (* Jean-François Alcover, Feb 08 2016, after Joerg Arndt *)
Table[1/n * Sum[EulerPhi[n/d] Total@ Map[Fibonacci, d + # & /@ {-1, 1}], {d, Divisors@ n}], {n, 47}] (* Michael De Vlieger, Dec 28 2016 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, EulerPhi[n/#] LucasL[#] &]/n]; (* Michael Somos, Jun 02 2019 *)
|
|
|
PROG
|
(PARI)
N=66; x='x+O('x^N);
B(x)=x*(1+x);
A=sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, eulerphi(n/d) * (fibonacci(d+1) + fibonacci(d-1)))/n)}; /* Michael Somos, Jun 02 2019 */
(Python)
from sympy import totient, lucas, divisors
def A000358(n): return (n&1^1)+sum(totient(n//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors(n, generator=True))//n # Chai Wah Wu, Sep 23 2023
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
