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A176500
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a(n) = 2*Farey(Fibonacci(n + 1)) - 3.
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13
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1, 3, 7, 19, 43, 115, 279, 719, 1879, 4911, 12659, 33235, 86715, 226315, 592767, 1551791, 4060203, 10630767, 27825227, 72843667, 190710291, 499271047, 1307051711, 3421933647, 8958716547, 23453948495, 61403187051, 160755514791, 420862602279, 1101832758583
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OFFSET
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1,2
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COMMENTS
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This sequence provides a 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 an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176502 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, Jul 12 2019]
The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 11 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. 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, 2Farey(m) - 3 = 43.
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MATHEMATICA
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a[n_] := 2 Sum[EulerPhi[k], {k, 1, Fibonacci[n+1]}] - 1;
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PROG
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(Magma) [2*(&+[EulerPhi(k):k in [1..Fibonacci(n+1)]])-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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