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

1,2

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

Table of n, a(n) for n=1..35.

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.

Jesus Omar Sistos Barron, Counting Conjugates of Colored Compositions, Honors College Thesis, Georgia Southern Univ. (2024), No. 985. See p. 30.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A064236 A339585 A245001 * A007569 A054317 A065840

Adjacent sequences: A365855 A365856 A365857 * A365859 A365860 A365861

Keyword

nonn

Author

Joshua P. Bowman, Sep 20 2023

Status

approved