A382431 Number of minimum dominating sets in the n-Goldberg graph.

63, 12, 5, 1395, 504, 204, 27, 5, 7370, 1728, 390, 42, 5, 21052, 3825, 621, 57, 5, 46011, 6930, 897, 72, 5, 86216, 11178, 1218, 87, 5, 146041, 16704, 1584, 102, 5, 230265, 23643, 1995, 117, 5, 344072, 32130, 2451, 132, 5, 493051, 42300, 2952, 147, 5, 683196, 54288, 3498, 162, 5

Offset: 3

Author

Eric W. Weisstein, Mar 25 2025

Keywords

• nonn
• easy

Comments

The domination number is given by A364668(n). - 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 Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,5,0,0,0,0,-10,0,0,0,0,10,0,0,0,0,-5,0,0,0,0,1).

Crossrefs

Cf. A364668 (domination number), A382384, A382657.

Programs

Mathematica

Table[Piecewise[{{1395, n == 6}, {5, Mod[n, 5] == 0}, {3 n, Mod[n, 5] == 4}, {3 n (3 n + 61)/10, Mod[n, 5] == 3}, {9 n (n^2 + 61 n - 76)/50, Mod[n, 5] == 2}, {n (27 n^3 + 3294 n^2 + 25281 n - 42602)/1000, Mod[n, 5] == 1}}], {n, 3, 50}] (* Eric W. Weisstein, May 31 2025 *)
Join[{63, 12, 5, 1395}, LinearRecurrence[{0, 0, 0, 0, 5, 0, 0, 0, 0, -10, 0, 0, 0, 0, 10, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1}, {504, 204, 27, 5, 7370, 1728, 390, 42, 5, 21052, 3825, 621, 57, 5, 46011, 6930, 897, 72, 5, 86216, 11178, 1218, 87, 5, 146041}, 50]] (* Eric W. Weisstein, May 31 2025 *)
CoefficientList[Series[-(1/((-1 + x)^5 (1 + x + x^2 + x^3 + x^4)^5)) (63 + 12 x + 5 x^2 + 1395 x^3 + 504 x^4 - 111 x^5 - 33 x^6 - 20 x^7 + 395 x^8 - 792 x^9 + 27 x^11 + 30 x^12 - 1848 x^13 + 225 x^14 + 81 x^15 - 3 x^16 - 20 x^17 + 501 x^18 + 45 x^19 - 33 x^20 - 3 x^21 + 5 x^22 - 44 x^23 + 18 x^24 + 6 x^28), {x, 0, 50}], x] (* Eric W. Weisstein, May 31 2025 *)

PARI

a(n)=my(m=n\5+1,r=-n%5); if(r<=2, if(r==0, 5, 3*n*if(r==1, 1, (3*m+11)/2)), if(n==6, 1395, n*if(r==3, 9*(m^2 + 11*m - 10)/2, (27*m^3 + 594*m^2 + 9*m - 742)/8) )) \\ Andrew Howroyd, May 24 2025

Formula

a(5*n) = 5.

a(5*n-1) = 3*(5*n-1); a(5*n-2) = 3*(5*n-2)*(3*n+11)/2; a(5*n-3) = 9*(5*n-3)*(n^2 + 11*n - 10)/2; a(5*n-4) = (5*n-4)*(27*n^3 + 594*n^2 + 9*n - 742)/8 for n > 2. - Andrew Howroyd, May 24 2025

Extensions

a(11)-a(15) from Eric W. Weisstein, May 12 2025

a(16) onwards from Andrew Howroyd, May 24 2025