cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Joshua P. Bowman, Sep 20 2023

Keywords

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.

Crossrefs

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