A365858 Number of cyclic compositions of 2*n-1 into odd parts.

1, 2, 3, 5, 10, 19, 41, 94, 211, 493, 1170, 2787, 6713, 16274, 39651, 97109, 238838, 589527, 1459961, 3626242, 9030451, 22542397, 56393862, 141358275, 354975433, 892893262, 2249412291, 5674891017, 14335757586, 36259245523, 91815545801, 232745229290, 590586152235, 1500020153485, 3813274653414

Offset: 1

Author

Joshua P. Bowman, Sep 20 2023

Keywords

• nonn

Comments

Odd bisection of A032189.

Also the number of cyclic compositions into an odd number of odd parts; because such a sum must be odd, alternating terms are zero and have been removed.

Links

• Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 25.
• Brian Hopkins, Jesús Sistos Barrón, and Hua Wang, Conjugating cyclic n-color compositions, Utilitas Mathematica (2025) Vol. 122, 53-64. See p. 62.
• Jesus Omar Sistos Barron, Counting Conjugates of Colored Compositions, Honors College Thesis, Georgia Southern Univ. (2024), No. 985. See p. 30.

Crossrefs

Cf. A000010, A000204 (Lucas), A032189, A365857, A365859.

Programs

Mathematica

Table[1/(2*n - 1) * Sum[EulerPhi[k]*LucasL[(2*n - 1)/k], {k, Divisors[2*n - 1]}], {n, 1, 40}] (* Vaclav Kotesovec, Sep 22 2023 *)

PARI

N=99;  x='x+O('x^N); B(x)=x/(1-x^2);
A=Vec(sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k)))));
vector(#A\2,n,A[2*n-1]) \\ Joerg Arndt, Sep 22 2023

Python

from sympy import totient, lucas, divisors
def A365858(n): return sum(totient(((n<<1)-1)//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors((n<<1)-1,generator=True))//((n<<1)-1) # Chai Wah Wu, Sep 23 2023

Formula

G.f.: (1/2) * Sum_{k odd} (phi(k)/k)*log((1+x^k-x^(2k))/(1-x^k-x^(2*k))), where phi(n) = A000010(n).

a(n) = (1/(2*n-1)) * Sum_{k divides 2n-1} phi(k)*A000204((2*n-1)/k).

a(n) ~ ((1+sqrt(5))/2)^(2*n-1) / (2*n). - Vaclav Kotesovec, Sep 22 2023