A304558 -id:A304558 - cp's OEIS frontend

A304553 Number of total dominating sets in the n-triangular honeycomb bishop graph.

0, 3, 29, 578, 22361, 1640001, 230504570, 62711805029, 33349745638405, 34939148029874116, 72519613634620549271, 299330052017563038619309, 2462757900094871658430161896, 40447266466705082294217088869999, 1327138203952448478777157651757710177, 87038459949342074077614931718690581743298

Offset: 1

Eric W. Weisstein, May 14 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Andrew Howroyd, PARI Program, May 2025.
Eric Weisstein's World of Mathematics, Total Dominating Set.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.

Cf. A290941, A304558, A304564.

vector(10,n,A304553(n)) \\ See PARI link. - Andrew Howroyd, May 16 2025

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

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

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

Original entry on oeis.org

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

Eric W. Weisstein, May 14 2018

nonn

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.

Cf. A304553, A304558, A382776, A382777.

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

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)

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

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

cp's OEIS Frontend

A304553 Number of total dominating sets in the n-triangular honeycomb bishop graph.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions