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
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).
Programs
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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
Comments