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A100107 Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,..

%I A100107 #15 Dec 10 2021 04:14:21
%S A100107 1,4,5,11,12,26,30,58,81,138,200,355,522,876,1380,2265,3572,5880,9350,
%T A100107 15272,24510,39806,64080,104084,167773,271968,439285,711530,1149852,
%U A100107 1862022,3010350,4873112,7881400,12755618,20633280,33391491,54018522,87413156
%N A100107 Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,..
%H A100107 T. D. Noe, <a href="/A100107/b100107.txt">Table of n, a(n) for n = 1..1000</a>
%F A100107 a(n) = Sum_{d|n} Lucas(d) = Sum_{d|n} A000032(d).
%F A100107 G.f.: Sum_{k>=1} Lucas(k) * x^k/(1 - x^k) = Sum_{k>=1} x^k * (1 + 2*x^k)/(1 - x^k - x^(2*k)). - _Ilya Gutkovskiy_, Aug 14 2019
%F A100107 a(n) ~ phi^n, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 10 2021
%e A100107 a(2) = 4 because the prime 2 is divisible only by 1 and 2, so L(1) + L(2) = 1 + 3 = 4.
%e A100107 a(3) = 5 because the prime 3 is divisible only by 1 and 3, so L(1) + L(3) = 1 + 4 = 5.
%e A100107 a(4) = 11 because the semiprime 4 is divisible only by 1, 2, 4, so L(1) + L(2) + L(4) = 1 + 3 + 7 = 11.
%p A100107 with(numtheory): with(combinat): a:=proc(n) local div: div:=divisors(n): sum(2*fibonacci(div[j]+1)-fibonacci(div[j]),j=1..tau(n)) end: seq(a(n),n=1..42); # _Emeric Deutsch_, Jul 31 2005
%t A100107 Table[Plus @@ Map[Function[d, LucasL[d]], Divisors[n]], {n, 100}] (* _T. D. Noe_, Aug 14 2012 *)
%Y A100107 Cf. A000032, A007435, A100279.
%K A100107 nonn
%O A100107 1,2
%A A100107 _Jonathan Vos Post_, Dec 26 2004
%E A100107 More terms from _Emeric Deutsch_, Jul 31 2005