A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).

1, -1, 2, -3, 3, -4, 8, -9, 6, -9, 14, -15, 16, -17, 27, -33, 21, -22, 36, -37, 34, -45, 61, -62, 51, -55, 73, -82, 76, -77, 124, -125, 80, -97, 120, -132, 132, -133, 171, -190, 153, -154, 221, -222, 194, -233, 296, -297, 239, -248, 313, -337, 301, -302

Offset

1,3

Links

Table of n, a(n) for n=1..54.

Formula

G.f. A(x) satisfies: A(x) = x * ( 1 - A(x) + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... ).

G.f.: x * ( 1 - Sum_{n>=1} a(n) * x^n / (1 + x^n) ).

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      add((-1)^((n-1)/d)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..54);  # Alois P. Heinz, Feb 10 2022

Mathematica

a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 54}]
nmax = 54; A[_] = 0; Do[A[x_] = x (1 + Sum[(-1)^k A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Crossrefs

Cf. A003238, A067856, A281487, A307776, A307778, A308077, A325144, A343370.

Sequence in context: A202560 A227165 A173933 * A193821 A130743 A263775

Adjacent sequences: A351404 A351405 A351406 * A351408 A351409 A351410

Keyword

sign

Author

Ilya Gutkovskiy, Feb 10 2022

Status

approved