A107392 Number of (inequivalent) fuzzy subgroups of the direct sum of group of integers modulo p^n and group of integers modulo 2 for a prime p with (p,2) = 1. Z_{p^n} + Z_2.
7, 31, 103, 303, 831, 2175, 5503, 13567, 32767, 77823, 182271, 421887, 966655, 2195455, 4947967, 11075583, 24641535, 54525951, 120061951, 263192575, 574619647, 1249902591, 2709520383, 5855248383, 12616466431, 27111981055, 58116276223, 124285616127
Offset: 0
Examples
a(3) = 303. A fuzzy subgroup is simply a chain of subgroups in the lattice of subgroups. Counting of chains in the lattice of subgroups of Z_{p^3} + Z_2 gives us a(3) = 303. The two papers cited describe the counting process using fuzzy subgroup concept.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- V. Murali, FSRG, Rhodes University.
- V. Murali and B. B. Makamba, Fuzzy subgroups of finite Abelian groups, Far East Journal of Mathematical Sciences (FJMS), Vol. 14, No. 1 (2004), pp. 113-125.
- V. Murali and B. B. Makamba, Counting the fuzzy subgroups of an Abelian group of order p^n q^m, Fuzzy Sets And Systems, Vol. 144, No.3 (2004), pp. 459-470.
- Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).
Programs
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Mathematica
LinearRecurrence[{7,-18,20,-8},{7,31,103,303},30] (* Harvey P. Dale, Dec 31 2015 *) -
PARI
Vec((12*x^2-18*x+7)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Jan 15 2015
Formula
a(n) = (2^n)*(n^2 + 7n + 8) - 1 for n=0..14.
G.f.: (12*x^2 - 18*x + 7) / ((x-1)*(2*x-1)^3). - Colin Barker, Jan 15 2015
Extensions
Corrected by T. D. Noe, Nov 08 2006
Comments