A176501
a(n) = Farey(m; I) where m = Fibonacci(n + 1) and I = [1/n, 1].
Original entry on oeis.org
1, 2, 4, 9, 19, 50, 122, 317, 837, 2213, 5758, 15236, 40028, 105079, 276627, 727409, 1910685, 5020094, 13180380, 34600740, 90814431, 238288480, 625111687, 1639676484, 4300183922, 11275936787, 29564497466, 77507123132, 203175049457, 532552499826, 1395790412496
Offset: 1
n = 5, I = [1/5, 1], m = Fibonacci(5 + 1) = 8, Farey(8) = 23, Farey(8; I) = 19
- Antoine Mathys, Table of n, a(n) for n = 1..40
- 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).
- Sameen Ahmed Khan, Mathematica notebook
- Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).
- Hugo Pfoertner, Ratio for networks with bridges, Plot2 of A174283(n)/a(n).
- Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).
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a[n_ /; n<4] := 2^(n-1); a[n_] := Module[{m = Fibonacci[n+1], v}, v = Reap[ Do[Sow[j/i], {i, n+1, m}, {j, 1, (i-1)/n}]][[2, 1]]; Total[ EulerPhi[ Range[m]]] - Length[v // Union]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* Jean-François Alcover, Aug 30 2018, after Antoine Mathys *)
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farey(n) = sum(i=1, n, eulerphi(i)) + 1;
a(n) = my(m=fibonacci(n + 1), count=0); for(b=n+1, m, for(a=1, (b-1)/n, if(gcd(a,b)==1, count++))); farey(m) - 1 - count; \\ Antoine Mathys, May 07 2019
A051389
Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.
Original entry on oeis.org
1, 2, 4, 8, 20, 42, 102, 250, 610, 1486, 3710, 9228, 23050, 57718, 145288, 365820, 922194, 2327914, 5885800, 14890796, 37701452, 95550472, 242325118, 614869792, 1561228066, 3966071764, 10080113232, 25630109268, 65194419268, 165890640468
Offset: 1
The a(1) = 1 resistance value is 1 ohm.
The a(2) = 2 resistance values are {1/2, 2}.
The a(3) = 4 resistance values are {1/3, 2/3, 3/2, 3}.
The a(4) = 8 resistance values are {1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4}.
The a(5) = 20 resistance values are {1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/6, 6/7, 7/6, 6/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, 5}.
E.g. 6/5 is made from two resistors in series in parallel with three resistors in series, since 6/5 = 1/(1/2 + 1/3). It cannot be obtained using fewer resistors.
Original entry on oeis.org
0, 1, 3, 7, 17, 41, 107, 287, 809, 2341, 6965, 21101, 65031, 202939, 640441, 2039509, 6546861, 21158437, 68791923, 224839127, 738316629, 2434622357, 8058616301, 26765349429, 89173526191, 297942766331, 998072479961, 3351459203873
Offset: 0
- Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
A176497
a(n) is the cardinality of the "Cross Set" which is the subset of distinct resistances that can be produced by a circuit of n unit resistors using only series or parallel combinations which cannot be decomposed as a single unit resistor in either series or parallel with a circuit of n-1 unit resistors.
Original entry on oeis.org
0, 0, 0, 1, 4, 9, 25, 75, 195, 475, 1265, 3135, 7983, 19697, 50003, 126163, 317629, 802945, 2035619, 5158039, 13084381, 33240845, 84478199, 214717585, 546235003, 1389896683, 3537930077, 9007910913, 22942258567, 58444273501
Offset: 1
A(1) has no cross set and the first term is defined to be zero; the cross sets for n = 2 and n = 3 are empty hence the second and third term are each zero. Noting that A(3) = 4 and A(4) = 9, the fourth term is 1. The fifth term is 4.
A340921
a(n) is the number of distinct resistances that can be produced using at most n unit resistors in a planar network.
Original entry on oeis.org
1, 2, 4, 8, 16, 36, 80, 194, 506, 1400, 4024, 11870, 35200, 104836, 311686, 929088, 2776618, 8321128, 24967712, 74985708
Offset: 0
A341536
Number of distinct resistances that can be produced using at most n unit resistors in series, parallel, bridge or fork configurations.
Original entry on oeis.org
1, 2, 4, 8, 16, 36, 80, 194, 500, 1342, 3623, 9835, 26412, 70505, 187805, 500627
Offset: 0
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