%I #14 Nov 18 2023 18:08:24

%S 1,1,8,15,50,99,232,429,835,1430,2480,3978,6372,9690,14640,21318,

%T 30789,43263,60280,82225,111254,148005,195416,254475,329095,420732,

%U 534496,672452,841160,1043460,1287648,1577532,1923465,2330445,2811240,3372291,4029178

%N Number of n element multisets of length 3 vectors over GF(2) that sum to zero.

%C a(n) is the number of n X 3 binary matrices under row permutations and column complementations.

%C See A362905 for other interpretations.

%H Andrew Howroyd, <a href="/A362906/b362906.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1).

%F G.f.: (1 - 3*x + 6*x^2 - 3*x^3 + x^4)/((1 - x)^8*(1 + x)^4).

%F a(n) = binomial(n+7, 7)/8 for odd n;

%F a(n) = (binomial(n+7, 7) + 7*binomial(n/2+3, 3))/8 for even n.

%e The a(1) = 1 multiset is {000}.

%e The a(2) = 8 multisets are {000, 000}, {001, 001}, {010, 010}, {011, 011}, {100, 100}, {101, 101}, {110, 110}, {111, 111}.

%e 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}.

%t 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 *)

%o (PARI) a(n) = (binomial(n+7,7) + if(n%2==0, 7*binomial(n/2+3, 3)))/8

%Y Column k=3 of A362905.

%Y Cf. A006381.

%K nonn,easy

%O 0,3

%A _Andrew Howroyd_, May 27 2023