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

0,1

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, n-r)* 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. 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).

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

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^2-18*x+7)/((x-1)*(2*x-1)^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