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
Keywords
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.
Programs
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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
Comments