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Partial sums of A006206.
2

%I #29 Apr 22 2020 18:30:19

%S 1,2,3,4,6,8,12,17,25,36,54,79,119,177,267,402,612,928,1420,2170,3334,

%T 5125,7911,12216,18926,29346,45610,70960,110610,172577,269685,421830,

%U 660648,1035603,1625123

%N Partial sums of A006206.

%H James Spahlinger, <a href="/A001461/b001461.txt">Table of n, a(n) for n = 1..1000</a>

%H D. J. Broadhurst, <a href="https://arxiv.org/abs/hep-th/9604128">On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory</a>, arXiv:hep-th/9604128, 1996.

%p b := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + mobius(n/d)*(fibonacci(d+1)+fibonacci(d-1)) od; RETURN(sum/n); end; A001461 := proc(n) local i; add(b(i),i=1..n); end;

%t b[n_] := Sum[MoebiusMu[n/d] (Fibonacci[d + 1] + Fibonacci[d - 1]), {d, Divisors[n]}]/n; Accumulate[Table[b[n], {n, 35}]] (* _Jean-François Alcover_, Dec 02 2011 *)

%o (Haskell)

%o a001461 n = a001461_list !! (n-1)

%o a001461_list = scanl1 (+) a006206_list -- _Reinhard Zumkeller_, Jun 01 2013

%o (Sage)

%o def a(n):

%o return sum((fibonacci(d + 1) + fibonacci(d - 1)) * moebius(n // d) for d in divisors(n)) // n

%o def b(n):

%o return sum(a(i) for i in range(1, n + 1))

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_, _David Broadhurst_