A107392 -id:A107392 - cp's OEIS frontend

A107464 Number of fuzzy subgroups of rank 3 cyclic group of order (p^n)qr 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

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

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.

			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.

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.

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).

Cf. A007047, A107392.

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

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

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

A107953 Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).

Original entry on oeis.org

31, 175, 703, 2415, 7551, 22143, 61951, 167167, 438271, 1122303, 2818047, 6959103, 16941055, 40730623, 96862207, 228130815, 532676607, 1234173951, 2839543807, 6491734015, 14755561471, 33361494015, 75061264383, 168124481535, 375004332031, 833223655423

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005

This sequence is one of a triple sequence A(n,m,l) of the number of fuzzy subsets of a set with n+m+l objects of 3 kinds. There are n,m and l objects for each kind respectively. Here a(n)= A(n,2,1). The sequence A107464 is one other example of A(n,m,l) for m=l=1.

			a(3) = 2^4*((9/2)*16 + 21*3 + 16) - 1 = 2415 which is the number of distinct chains in the power set lattice (or fuzzy subsets) of a set X_(n+3) with 3 kinds of objects, n of one kind, 2 of another and one of yet another.

V. Murali, On the number of fuzzy subsets of an (n+3)-element set of specification n^1 2^1 1, Rhodes University Preprint, 2005.

Colin Barker, Table of n, a(n) for n = 0..1000
Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A107464, A007047, A107392.

Table[2^(n+1) (n^2/2 (n+13)+21n+16)-1,{n,0,30}] (* or *) LinearRecurrence[ {9,-32,56,-48,16},{31,175,703,2415,7551},30] (* Harvey P. Dale, Feb 10 2015 *)

Vec((48*x^3-120*x^2+104*x-31)/((x-1)*(2*x-1)^4) + O(x^100)) \\ Colin Barker, Jan 15 2015

a(n) = 2^(n+1) * ((n^2/2)*(n+13) + 21*n + 16) - 1.
G.f.: (48*x^3-120*x^2+104*x-31) / ((x-1)*(2*x-1)^4). - Colin Barker, Jan 15 2015
a(0)=31, a(1)=175, a(2)=703, a(3)=2415, a(4)=7551, a(n)=9*a(n-1)- 32*a(n-2)+ 56*a(n-3)-48*a(n-4)+16*a(n-5). - Harvey P. Dale, Feb 10 2015

a(5) corrected Jun 01 2005

Incorrect term deleted by Colin Barker, Jan 15 2015

A107954 Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.

Original entry on oeis.org

79, 527, 2415, 9263, 31871, 101759, 307455, 890111, 2490367, 6774783, 18001919, 46886911, 120029183, 302678015, 753205247, 1852375039, 4507828223, 10866393087, 25970081791, 61583917055, 144997089279, 339159810047

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 30 2005

This sequence is an example of another line in a triple sequence A(n,m,l) with n a nonnegative integer, m = 2 and l = 1. It is related to sequences A107464, A107953 which are part of the same triple sequence with different parameter values for m and l.

			a(2) = 8 * ( (16 + 184)/6 + (316 + 370)/3 + 40 ) - 1 = 2415. This is the number of fuzzy subsets of a set of (2+4) elements of which 2 are of one kind, 3 are of another kind and 1 of a kind distinct from the other two.

V. Murali, On the enumeration of fuzzy subsets of X_(n+4) of specification n^1 3^1 1, Rhodes University JRC-Abstract-Report, In Preparation, 12 pages 2005.

Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (11,-50,120,-160,112,-32).

Cf. A007047, A107392, A107464, A107953.

a[n_] := 2^n(n^4 + 23n^3 + 158n^2 + 370n + 240)/3 - 1; Table[ a[n], {n, 0, 21}] (* Robert G. Wilson v, May 31 2005 *)
LinearRecurrence[{11,-50,120,-160,112,-32},{79,527,2415,9263,31871,101759},40] (* Harvey P. Dale, Aug 15 2025 *)

a(n) = 2^(n+1)*( (n^4 + 23*n^3)/6 + (79*n^2 + 185*n)/3 + 40 ) - 1.

G.f.: (128*x^4-432*x^3+568*x^2-342*x+79) / ((x-1)*(2*x-1)^5). [Colin Barker, Dec 10 2012]

a(6)-a(21) from Robert G. Wilson v, May 31 2005

A107955 Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.

Original entry on oeis.org

191, 1471, 7551, 31871, 119231, 410303, 1327103, 4090623, 12130303, 34842623, 97435647, 266313727, 713637887, 1879523327, 4875091967, 12474187775, 31531728895, 78832992255, 195135799295, 478649778175, 1164351373311

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), Jun 01 2005

This sequence is another example, together with A107953 and A107954, of a triple sequence A(n,m,l) with n a nonnegative integer, m = 4 and l = 1.

			a(3) = (2^(3+1))*(1/24)*(3^5 + 36 * 3^4 + 431 * 3^3 + 2088 * 3^2 + 3972 * 3 + 2304) - 1 = 31871. This is the number of chains in the power set lattice (which is also the number of fuzzy subsets) of X_(n+5).

Venkat Murali, On the enumeration of fuzzy subsets of an (n+5)-element set X_(n+5) of specification n^1 4^1 1, Rhodes University JRC-Abstract-Report, In Preparation, 15 pages 2005.

Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (13,-72,220,-400,432,-256,64).

Cf. A007047, A107392, A107464, A107953, A107954.

a(n) = (2^(n+1))*(1/24)*(n^5 + 36*n^4 + 431*n^3 + 2088*n^2 + 3972*n + 2304) - 1,

G.f.: (320*x^5-1360*x^4+2400*x^3-2180*x^2+1012*x-191) / ((x-1)*(2*x-1)^6). [Colin Barker, Dec 10 2012]

cp's OEIS Frontend

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.

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A107953 Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).

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A107954 Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.

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A107955 Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.

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Examples

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Formula

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