OFFSET
0,3
COMMENTS
a(n) is the number of n X 3 binary matrices under row permutations and column complementations.
See A362905 for other interpretations.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1).
FORMULA
G.f.: (1 - 3*x + 6*x^2 - 3*x^3 + x^4)/((1 - x)^8*(1 + x)^4).
a(n) = binomial(n+7, 7)/8 for odd n;
a(n) = (binomial(n+7, 7) + 7*binomial(n/2+3, 3))/8 for even n.
EXAMPLE
The a(1) = 1 multiset is {000}.
The a(2) = 8 multisets are {000, 000}, {001, 001}, {010, 010}, {011, 011}, {100, 100}, {101, 101}, {110, 110}, {111, 111}.
The a(3) = 15 multisets are {000, 000, 000}, {000, 001, 001}, {000, 010, 010}, {000, 011, 011}, {000, 100, 100}, {000, 101, 101}, {000, 110, 110}, {000, 111, 111}, {001, 010, 011}, {001, 100, 101}, {001, 110, 111}, {010, 100, 110}, {010, 101, 111}, {011, 100, 111}, {011, 101, 110}.
MATHEMATICA
A362906[n_]:=(Binomial[n+7, 7]+If[EvenQ[n], 7Binomial[n/2+3, 3], 0])/8; Array[A362906, 50, 0] (* Paolo Xausa, Nov 18 2023 *)
PROG
(PARI) a(n) = (binomial(n+7, 7) + if(n%2==0, 7*binomial(n/2+3, 3)))/8
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrew Howroyd, May 27 2023
STATUS
approved