A382657 Number of minimum total dominating sets in the n-Goldberg graph.

16, 277, 10, 2386, 28, 33301, 360, 10, 4334, 60, 67288, 728, 10, 9856, 102, 150750, 1292, 10, 19222, 154, 299368, 2124, 10, 34112, 216, 549276, 3306, 10, 56730, 288, 951456, 4930, 10, 89916, 370, 1576202, 7098, 10, 137268, 462, 2518596, 9922, 10, 203274, 564, 3905148, 13524, 10

Offset: 3

Author

Eric W. Weisstein, Apr 02 2025

Keywords

• nonn
• easy

Comments

For n > 3, the total domination number is given by 2*floor((7*n+4)/5) = 2*A047332(n+1). For n = 3, the total domination number is 9. - Andrew Howroyd, May 24 2025

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..1000
• Eric Weisstein's World of Mathematics, Goldberg Graph.
• Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,9, 0,0,0,0,-36, 0,0,0,0,84, 0,0,0,0,-126, 0,0,0,0,126, 0,0,0,0,-84, 0,0,0,0,36, 0,0,0,0,-9, 0,0,0,0,1).

Crossrefs

Cf. A382431.

Programs

PARI

a(n) = { my(m=n\5+1,r=-n%5); if(n<=8, [16, 277, 10, 2386, 28, 33301][n-2], if(r<=2, if(r==0, 10, n*if(r==1, (m + 13)*(m^2 + 2*m + 24)/12, (m^7 + 70*m^6 + 1750*m^5 + 39340*m^4 + 525889*m^3 + 2944270*m^2 + 15922920*m + 12216960)/20160)), n*if(r==3, (m + 2), (m^5 + 35*m^4 + 365*m^3 + 5485*m^2 + 21114*m + 16200)/360) )) } \\ Andrew Howroyd, May 24 2025

Formula

a(5*n) = 10.

From Andrew Howroyd, May 24 2025: (Start)

a(5*n-1) = (5*n-1)*(n + 13)*(n^2 + 2*n + 24)/12 for n >= 2;

a(5*n-2) = (5*n-2)*(n^7 + 70*n^6 + 1750*n^5 + 39340*n^4 + 525889*n^3 + 2944270*n^2 + 15922920*n + 12216960)/20160 for n >= 3;

a(5*n-3) = (5*n-3)*(n + 2) for n >= 2;

a(5*n-4) = (5*n-4)*(n^5 + 35*n^4 + 365*n^3 + 5485*n^2 + 21114*n + 16200)/360 for n >= 3. (End)

Extensions

a(8)-a(12) from Eric W. Weisstein, May 11 2025

a(13) onwards from Andrew Howroyd, May 24 2025