cp's OEIS Frontend

A281487 a(n+1) = -Sum_{d|n} a(d), a(1) = 1.

%I A281487 #17 Apr 29 2019 17:23:03
%S A281487 1,-1,0,-1,1,-2,2,-3,4,-5,4,-5,8,-9,7,-9,13,-14,12,-13,18,-21,17,-18,
%T A281487 29,-31,23,-28,36,-37,36,-37,50,-55,42,-46,64,-65,53,-62,83,-84,75,
%U A281487 -76,94,-107,90,-91,129,-132,107,-121,145,-146,135,-141,180,-193,157
%N A281487 a(n+1) = -Sum_{d|n} a(d), a(1) = 1.
%C A281487 a(1) = 1, any other choice simply adds a factor to all terms.
%C A281487 Observations: sign of a(n) is -(-1)^n, the subsequences |a(n)| with n = 1, 2 mod 4 and |a(n)| with n = 3, 0 mod 4 both grow at n>5. Both these subsequences seem to share the asymptotics with A003238 (and hence A000123): log(|a(n)|) is approximately proportional to (log(n/log(n)))^2; however, the factor is much less than log(4).
%C A281487 There is a family of sequences with the formula a(n) = s*Sum_{d|(n-k), 1<=d<n} a(d). For s=+1 and k = 0,1,2, these are A002033, A003238, A007439. For s=-1 and k = 0,1,2, these are the Möbius function A008683, this sequence, and A281488.
%H A281487 Andrey Zabolotskiy, <a href="/A281487/b281487.txt">Table of n, a(n) for n = 1..20000</a>
%F A281487 a(1) = 1.
%F A281487 a(n+1) = -Sum_{d|n} a(d) for n>=1.
%F A281487 a(n+1) = Sum_{d|n} |a(d)|*(-1)^(d+n) for n>=1.
%F A281487 From _Ilya Gutkovskiy_, Apr 29 2019: (Start)
%F A281487 G.f.: x * (1 - Sum_{n>=1} a(n)*x^n/(1 - x^n)).
%F A281487 L.g.f.: log(Product_{n>=1} (1 - x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n. (End)
%e A281487 a(9) = -(a(1)+a(2)+a(4)+a(8)) = -(1-1-1-3) = 4.
%o A281487 (Python)
%o A281487 a = [1]
%o A281487 for n in range(1, 100):
%o A281487    a.append(-sum(a[d-1] for d in range(1, n+1) if n%d == 0))
%o A281487 print(a)
%o A281487 (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n-1, d, va[d]);); va;} \\ _Michel Marcus_, Apr 29 2019
%Y A281487 Cf. A003238, A281488, A000123.
%K A281487 sign,easy
%O A281487 1,6
%A A281487 _Andrey Zabolotskiy_, Jan 22 2017