

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.


5



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
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OFFSET

0,1


COMMENTS

This is just one row of a double sequence a(n,m) for n = 0,1,2, ... and m = 0,1,2,...: a(n,m) = 2^(n+m+1)*(Sum_{r=0..m} (2^(r) * binomial(n, nr)* binomial(m, r)))  1, with 0 <= m <= n and a(0,0)=1.


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. 113125.
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. 459470.
Index entries for linear recurrences with constant coefficients, signature (7,18,20,8).


FORMULA

a(n) = (2^n)*(n^2 + 7n + 8)  1 for n=0..14.
G.f.: (12*x^2  18*x + 7) / ((x1)*(2*x1)^3).  Colin Barker, Jan 15 2015


EXAMPLE

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.


MATHEMATICA

LinearRecurrence[{7, 18, 20, 8}, {7, 31, 103, 303}, 30] (* Harvey P. Dale, Dec 31 2015 *)


PROG

(PARI) Vec((12*x^218*x+7)/((x1)*(2*x1)^3) + O(x^100)) \\ Colin Barker, Jan 15 2015


CROSSREFS

Sequence in context: A218956 A139876 A222265 * A054497 A235593 A119359
Adjacent sequences: A107389 A107390 A107391 * A107393 A107394 A107395


KEYWORD

nonn,easy


AUTHOR

Venkat Murali (v.murali(AT)ru.ac.za), May 25 2005


EXTENSIONS

Corrected by T. D. Noe, Nov 08 2006


STATUS

approved



