A319777 a(n) is the number of equivalence classes of triples of sets each with n or fewer elements where two triples are equivalent if the number of elements in all intersections is the same.

1, 15, 100, 436, 1459, 4069, 9929, 21871, 44426, 84494, 152171, 261749, 432906, 692102, 1074198, 1624314, 2399943, 3473337, 4934182, 6892578, 9482341, 12864643, 17232007, 22812673, 29875352, 38734384, 49755317, 63360923, 80037668, 100342652, 124911036

Offset

0,2

Comments

A019298(n) is the analogous sequence if the three sets must each have exactly n elements.

Links

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Formula

a(n) = Sum_{k=0..n} A244865(k). [corrected by Michel Marcus, Dec 27 2018]

From Colin Barker, Dec 27 2018: (Start)

G.f.: (1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^8*(1 + x)).

a(n) = 7*a(n-1) - 20*a(n-2) + 28*a(n-3) - 14*a(n-4) - 14*a(n-5) + 28*a(n-6) - 20*a(n-7) + 7*a(n-8) - a(n-9) for n>8.

(End)

Example

The triple (A, B, C) = ({1, 2}, {1, 2, 3}, {1, 4}) is equivalent to the triple (A', B', C') = ({1, 8}, {1, 4, 8}, {5, 8}) because all intersections of the sets in a triple are equal:
|A|         = |{1, 2}|    = 2 = |{1, 8}|    = |A'|
|B|         = |{1, 2, 3}| = 3 = |{1, 4, 8}| = |B'|
|C|         = |{1, 4}|    = 2 = |{5, 8}|    = |C'|
|A & B|     = |{1, 2}|    = 2 = |{1, 8}|    = |A' & B'|
|A & C|     = |{1}|       = 1 = |{8}|       = |A' & C'|
|B & C|     = |{1}|       = 1 = |{8}|       = |B' & C'|
|A & B & C| = |{1}|       = 1 = |{8}|       = |A' & B' & C'|

Maple

a:=n->add((15*(127+(-1)^k)+6432*k+8936*k^2+6480*k^3+2570*k^4+528*k^5+44*k^6)/1920, k=0..n): seq(a(n), n=0..30); # Muniru A Asiru, Sep 28 2018

Prog

(GAP)  List([0..30], n->Sum([0..n], k->(15*(127+(-1)^k)+6432*k+8936*k^2+6480*k^3+2570*k^4+528*k^5+44*k^6)/1920)); # Muniru A Asiru, Sep 28 2018
(PARI) a(n) = sum(k=0, n, (15*(127+(-1)^k) + 6432*k + 8936*k^2 + 6480*k^3 + 2570*k^4 + 528*k^5 + 44*k^6) / 1920); \\ Michel Marcus, Dec 27 2018
(PARI) Vec((1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^8*(1 + x)) + O(x^40)) \\ Colin Barker, Dec 28 2018

Crossrefs

Cf. A019298, A244865.

Cf. A000330(n-1) is analogous, but with pairs instead of triples.

Sequence in context: A000973 A034266 A087661 * A242657 A163717 A111370

Adjacent sequences: A319774 A319775 A319776 * A319778 A319779 A319780

Keyword

nonn

Author

Peter Kagey, Sep 26 2018

Status

approved