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

Author

Eric W. Weisstein, May 14 2018

Keywords

• nonn

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.

Crossrefs

Cf. A304553, A304558, A382776, A382777.

Programs

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

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)

Extensions

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

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