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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]
Farey(n) = A005728(n). - Franklin T. Adams-Watters, May 12 2010
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LINKS
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Antoine Mathys, Table of n, a(n) for n = 1..50
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (Apr 20 2010).
Sameen Ahmed KHAN, Mathematica notebook 1
Sameen Ahmed KHAN, Mathematica notebook 2
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FORMULA
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a(n) = 2 * A176499(n) - 3.
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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;
Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* Jean-François Alcover, Nov 03 2018, from PARI *)
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PROG
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(PARI) a(n) = 2*sum(k=1, fibonacci(n+1), eulerphi(k))-1 \\ Charles R Greathouse IV, Oct 07 2016
(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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Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176501, A176502.
Sequence in context: A192301 A055622 A075900 * A136041 A146685 A146653
Adjacent sequences: A176497 A176498 A176499 * A176501 A176502 A176503
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KEYWORD
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nonn
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AUTHOR
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Sameen Ahmed Khan, Apr 21 2010
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EXTENSIONS
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a(26)-a(28) from Sameen Ahmed Khan, May 02 2010
a(29)-a(30) from Antoine Mathys, Aug 06 2018
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STATUS
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approved
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