A176498 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 unit resistors in series and/or parallel.

0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 9, 24, 58, 124, 312, 759, 1768, 4421, 10811, 27191, 68591, 174627, 441633, 1124795, 2866004, 7297500, 18585359, 47337643, 120562041, 307063757

Offset: 1

Author

Sameen Ahmed Khan, Apr 21 2010

Keywords

• more
• nonn

Comments

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.

Examples

			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, ....
		

Links

• 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], (20 April 2010).
• S. A. Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - _N. J. A. Sloane_, Oct 23 2012

Crossrefs

Cf. A048211, A176497.

Extensions

a(16)-a(25) from Antoine Mathys, Mar 20 2017

a(26)-a(30) from Antoine Mathys, Dec 08 2024