

A176500


a(n) = 2*Farey(Fibonacci(n + 1))  3.


13



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

1,2


COMMENTS

This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or nonplanar) 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. AdamsWatters, May 12 2010
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 nonplanar 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


LINKS

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), 175179 (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.genph], (Apr 20 2010).
Sameen Ahmed KHAN, Mathematica notebook 1
Sameen Ahmed KHAN, Mathematica notebook 2
Hugo Pfoertner, Ratio for seriesparallel networks, Plot2 of A048211(n)/a(n).
Hugo Pfoertner, Ratio for planar networks with generalized bridges, Plot2 of A337516(n)/a(n).
Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).


FORMULA

a(n) = 2 * A176499(n)  3.


EXAMPLE

n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23, 2Farey(m)  3 = 43.


MATHEMATICA

a[n_] := 2 Sum[EulerPhi[k], {k, 1, Fibonacci[n+1]}]  1;
Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* JeanFrançois Alcover, Nov 03 2018, from PARI *)


PROG

(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


CROSSREFS

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176501, A176502.
Sequence in context: A192301 A055622 A075900 * A334099 A136041 A146685
Adjacent sequences: A176497 A176498 A176499 * A176501 A176502 A176503


KEYWORD

nonn


AUTHOR

Sameen Ahmed Khan, Apr 21 2010


EXTENSIONS

a(26)a(28) from Sameen Ahmed Khan, May 02 2010
a(29)a(30) from Antoine Mathys, Aug 06 2018


STATUS

approved



