A304564 Number of minimum total dominating sets in the n-triangular honeycomb bishop graph.

0, 2, 2, 6, 75, 21, 208, 3950, 540, 11220, 314880, 25740, 917280, 36029700, 1965600, 107100000, 5627890800, 219769200, 16995484800, 1153034190000, 33844456800, 3525796058400, 300234909744000, 6868433880000, 927359072640000, 96883959332160000, 1776393899280000, 301733192320560000

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.

Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.

Formula

From Andrew Howroyd, Apr 04 2025: (Start)

a(3*n) = A382777(n).

a(3*n+4) = Sum_{k=0..n} A382776(n,k)*(4*binomial(n+k+2,2) * binomial(2*n-k+2,2) + 2*binomial(n+k+3,3) * (2*n-k+1)).

See the PARI program for a(3*n+2). (End)

Prog

(PARI)
T(n, k)=binomial(2*n-k, k)*binomial(n+k, n-k)*(2*(n-k))!*(2*k)!/(2^n)
b1(n) = sum(k=0, n, T(n, k))
b2(n) = sum(k=0, n, T(n, k)*(2*binomial(n+k+3, 3)*(2*n-k+1) + 4*binomial(n+k+2, 2)*binomial(2*n-k+2, 2)))
b3(n) = sum(k=0, n, T(n, k)*(n+k)*(n+k+1)*(7*n-2*k+5)/3)
b4(n) = sum(k=0, n, T(n, k)*(2*binomial(n+k+4, 4)*(2*n-k+1) + 24*binomial(n+k+2, 2)*binomial(2*n-k+3, 3)))
b5(n) = sum(k=0, n, T(n, k)*(40*binomial(n+k+6, 6)*binomial(2*n-k+2, 2) + 240*binomial(n+k+5, 5)*binomial(2*n-k+3, 3) + 304*binomial(n+k+4, 4)*binomial(2*n-k+4, 4)))
a(n) = my(t=n\3); if(n%3==0, b1(t), if(n%3==1, b2(t-1), b1(t+1) + b3(t) + b4(t-1) + b5(t-2))) \\ Andrew Howroyd, Apr 09 2025

Crossrefs

Cf. A304553, A304558, A382776, A382777.

Sequence in context: A184712 A303225 A326943 * A181265 A382666 A093909

Adjacent sequences: A304561 A304562 A304563 * A304565 A304566 A304567

Keyword

nonn

Author

Eric W. Weisstein, May 14 2018

Extensions

a(8)-a(10) from Andrew Howroyd, May 19 2018

a(11) onwards from Andrew Howroyd, May 16 2025

Status

approved