A107464 Number of fuzzy subgroups of rank 3 cyclic group of order (p^n)*q*r where p, q and r are three distinct prime.

11, 51, 175, 527, 1471, 3903, 9983, 24831, 60415, 144383, 339967, 790527, 1818623, 4145151, 9371647, 21037055, 46923775, 104071167, 229638143, 504365055, 1103101951, 2403336191, 5217714175, 11291066367, 24360517631, 52412022783, 112474456063, 240786604031

Offset: 0

Author

Venkat Murali (v.murali(AT)ru.ac.za), May 27 2005

Keywords

• easy
• nonn

Comments

It would be good to find a formula for a(n,m,l) or generating function for the number of chains in the lattice of subgroups ( these are the fuzzy subgroups )of the direct sum Z_(p^n) + Z_(q^m) + Z_(r^l) for given 3 distinct prime p,q and r and for integers n,m and l.

Examples

			a(5) = (2^6)*(5^2+6*5+6)-1= 3903. This is the number of chains in the lattice of subgroups of the direct sum Z_(p^6)+ Z_q + Z_r for 3 distinct prime p,q and r where Z_i is the group of integers modulo i.
		

References

• V. Murali, Number of chains in the power set of a set with (n+2) elements, specification n^1 1^2, preprint, 2005.
• V. Murali and B. B. Makamba, Fuzzy subgroups of finite Abelian groups III, Rhodes University Preprint, 2005.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• V. Murali, FSRG, Rhodes University.
• Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Crossrefs

Cf. A007047, A107392.

Programs

PARI

Vec((16*x^2-26*x+11)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Jan 15 2015

Formula

a(n) = 2^(n+1)*(n^2 + 6n + 6) - 1.

G.f.: (16*x^2-26*x+11) / ((x-1)*(2*x-1)^3). - Colin Barker, Jan 15 2015

Extensions

Missing a(8) inserted by Colin Barker, Jan 15 2015