A382431 -id:A382431 - cp's OEIS frontend

A364668 Domination and lower independence number of the n-Goldberg graph.

0, 3, 5, 7, 9, 11, 14, 16, 18, 20, 22, 25, 27, 29, 31, 33, 36, 38, 40, 42, 44, 47, 49, 51, 53, 55, 58, 60, 62, 64, 66, 69, 71, 73, 75, 77, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 124, 126, 128, 130, 132

Offset: 0

Eric W. Weisstein, Aug 01 2023

The Goldberg graph is defined for n >= 3.
Extended to n = 0 through 2 using the formula/recurrence.
Disagrees with A195167(n) at n = 26, 31, 36, 41, ....

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Goldberg Graph.
Eric Weisstein's World of Mathematics, Lower Independence Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A382431.

Table[(11 n - Cos[2 n Pi/5] - Cos[4 n Pi/5] + Sqrt[1 + 2/Sqrt[5]] Sin[2 n Pi/5] + Sqrt[1 - 2/Sqrt[5]] Sin[4 n Pi/5] + 2)/5, {n, 0, 20}]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 3, 5, 7, 9, 11}, 20]
CoefficientList[Series[x (3 + 2 x + 2 x^2 + 2 x^3 + 2 x^4)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4)), {x, 0, 20}], x]

a(n) = a(n-1) + a(n-5) - a(n-6).
G.f.: x*(3+2*x+2*x^2+2*x^3+2*x^4)/((-1+x)^2*(1+x+x^2+x^3+x^4)).
a(n) = floor((11*n + 4)/5). - Andrew Howroyd, May 25 2025

Name extended by Eric W. Weisstein, Mar 10 2025

A382384 Number of minimum connected dominating sets in the n-Goldberg graph.

Original entry on oeis.org

6, 96, 290, 744, 1974, 5376, 15642, 45480, 124014, 343008, 944658, 2596776, 7116390, 19409664, 52694730, 142812648, 385840030, 1039911520, 2796034626, 7501233256, 20084164374, 53677896192, 143214557050, 381504047912, 1014784646094, 2695617288672, 7151420301682

Offset: 3

Eric W. Weisstein, Mar 23 2025

The connected domination number is given by 4*n - 1 = A004767(n - 1). - Andrew Howroyd, May 25 2025

Andrew Howroyd, Table of n, a(n) for n = 3..500
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Goldberg Graph.
Index entries for linear recurrences with constant coefficients, signature (4,2,-16,-7,20,24,-64,97,236,-246,-368,7,252,-772,-64,1920,0,-1024).

Cf. A004767, A382431.

LinearRecurrence[{4, 2, -16, -7, 20, 24, -64, 97, 236, -246, -368, 7, 252, -772, -64, 1920, 0, -1024}, {6, 96, 290, 744, 1974, 5376, 15642, 45480, 124014, 343008, 944658, 2596776, 7116390, 19409664, 52694730, 142812648, 385840030, 1039911520}, 20] (* Eric W. Weisstein, Jun 04 2025 *)
CoefficientList[Series[2 (3 + 36 x - 53 x^2 - 256 x^3 - 2 x^4 + 592 x^5 + 1030 x^6 + 616 x^7 - 2817 x^8 - 2804 x^9 + 2591 x^10 + 2200 x^11 - 2592 x^12 - 2176 x^13 + 5168 x^14 + 3840 x^15 - 2304 x^16 - 2048 x^17)/((1 - x)^2 (1 + x)^2 (1 - x - 4 x^2)^2 (1 - x + 2 x^2)^2 (1 - x^2 - 4 x^3)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 04 2025 *)

G.f.: 2*x^3*(3 + 36*x - 53*x^2 - 256*x^3 - 2*x^4 + 592*x^5 + 1030*x^6 + 616*x^7 - 2817*x^8 - 2804*x^9 + 2591*x^10 + 2200*x^11 - 2592*x^12 - 2176*x^13 + 5168*x^14 + 3840*x^15 - 2304*x^16 - 2048*x^17)/((1 - x)*(1 + x)*(1 - x + 2*x^2)*(1 - x - 4*x^2)*(1 - x^2 - 4*x^3))^2. - Andrew Howroyd, May 25 2025

a(7) onwards from Andrew Howroyd, May 24 2025

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

Original entry on oeis.org

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

Eric W. Weisstein, Apr 02 2025

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

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).

Cf. A382431.

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

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)

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

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

cp's OEIS Frontend

A364668 Domination and lower independence number of the n-Goldberg graph.

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A382384 Number of minimum connected dominating sets in the n-Goldberg graph.

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

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