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A176499
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Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).
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13
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2, 3, 5, 11, 23, 59, 141, 361, 941, 2457, 6331, 16619, 43359, 113159, 296385, 775897, 2030103, 5315385, 13912615, 36421835, 95355147, 249635525, 653525857, 1710966825, 4479358275, 11726974249, 30701593527, 80377757397, 210431301141, 550916379293
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OFFSET
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1,1
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COMMENTS
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This sequence arises in the analytically obtained strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides a strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176501 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]
The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is wrong. It only applies to purely serial-parallel networks (A048211), but it already fails when bridges are allowed, as described in A174283. Even more so if arbitrary nonplanar networks are allowed as in A337517. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 24 2021
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LINKS
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FORMULA
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EXAMPLE
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n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23.
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MAPLE
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with(numtheory): with(combinat, fibonacci): a:=n->1+add(phi(i), i=1..n): seq(a(fibonacci(n+1)), n=1..30); # Muniru A Asiru, Jul 31 2018
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MATHEMATICA
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b[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
a[n_] := b[Fibonacci[n + 1]];
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PROG
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(PARI) farey(n) = 1+sum(k=1, n, eulerphi(k));
(GAP) List([1..30], n->Sum([1..Fibonacci(n+1)], i->Phi(i)))+1; # Muniru A Asiru, Jul 31 2018
(Magma) [1+&+[EulerPhi(i):i in [1..Fibonacci(n+1)]]:n in [1..30]]; // Marius A. Burtea, Jul 26 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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