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
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
KEYWORD
nonn
AUTHOR
Joshua P. Bowman, Sep 20 2023
STATUS
approved