A176498 - OEIS

%I #28 Nov 20 2024 09:01:07

%S 0,0,0,0,0,0,0,0,1,6,9,24,58,124,312,759,1768,4421,10811,27191,68591,

%T 174627,441633,1124795,2866004,7297500,18585359,47337643

%N Number of elements less than 1/2 in the Cross Set which is the subset of the set of distinct resistances that can be produced using n equal resistors in series and/or parallel.

%C This sequence arises in the decomposition of the sets A(n + 1) of equivalent resistances, when n equal resistors are combined in series/parallel, into series parallel and cross sets respectively. All the elements of the parallel set are strictly less than 1 and all those of the series set are strictly greater than 1. The cross set is expected to be dense around 1 with very few elements below 1/2. Hence it is relevant to count the elements below 1/2.

%H Antoni Amengual, <a href="http://dx.doi.org/10.1119/1.19396">The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel</a>, American Journal of Physics, 68(2), 175-179 (February 2000).

%H Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1004.3346/">The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel</a>, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).

%H S. A. Khan, <a href="http://www.ias.ac.in/article/fulltext/pmsc/122/02/0153-0162">Farey sequences and resistor networks</a>, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - _N. J. A. Sloane_, Oct 23 2012

%e The order of the cross set is given by A176497: 0, 0, 0, 1, 4, 9, 25, 75, 195, 475, 1265, 3135, ... The sets corresponding n = 4 to n = 8 do not have a single element below 1/2. For n = 9 onwards we have a few elements which are less than 1/2; they are 1, 6, 9, 24, ....

%Y Cf. A048211, A176497.

%K more,nonn,changed

%O 1,10

%A _Sameen Ahmed Khan_, Apr 21 2010

%E a(16)-a(25) from _Antoine Mathys_, Mar 20 2017

%E a(26)-a(28) from _Antoine Mathys_, Nov 20 2024