A302759 Number of total dominating sets in the n-Andrásfai graph.
1, 11, 131, 1365, 12883, 113935, 967455, 8013983, 65410751, 529283583, 4261449727, 34213027327, 274240586751, 2196272295935, 17580376055807, 140687025184767, 1125685164621823, 9006288735567871, 72053745778425855, 576444534576513023, 4611617848860868607
Offset: 1
Links
- Eric Weisstein's World of Mathematics, Andrásfai Graph
- Eric Weisstein's World of Mathematics, Total Dominating Set
- Index entries for linear recurrences with constant coefficients, signature (23,-210,996,-2664,4032,-3200,1024).
Programs
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Mathematica
Join[{1}, Table[(8^(n + 1) + (2^n (n - 2) - 4^(n + 1) ) (3 n - 1) - 16)/16, {n, 2, 20}]] Join[{1}, LinearRecurrence[{23, -210, 996, -2664, 4032, -3200, 1024}, {11, 131, 1365, 12883, 113935, 967455, 8013983}, 20]] CoefficientList[Series[(-1 + 12 x - 88 x^2 + 334 x^3 - 706 x^4 + 928 x^5 - 672 x^6 + 256 x^7)/((-1 + 2 x)^3 (-1 + 4 x)^2 (1 - 9 x + 8 x^2)), {x, 0, 20}], x]
Formula
a(n) = (8^(n + 1) + (2^n*(n - 2) - 4^(n + 1))*(3*n - 1))/16 - 1 for n > 1.
a(n) = 23*a(n-1) - 210*a(n-2) + 996*a(n-3) - 2664*a(n-4) + 4032*a(n-5) - 3200*a(n-6) + 1024*a(n-7) for n > 8.
G.f.: x*(-1 + 12*x - 88*x^2 + 334*x^3 - 706*x^4 + 928*x^5 - 672*x^6 + 256*x^7)/((-1 + 2*x)^3*(-1 + 4*x)^2*(1 - 9*x + 8*x^2)).
E.g.f.: (1 - 8*exp(x) + 4*exp(8*x) + exp(4*x)*(2 - 24*x) + 2*x + exp(2*x)*(1 - 4*x + 6*x^2))/8. - Stefano Spezia, Aug 29 2025
Extensions
a(9)-a(21) from Andrew Howroyd, Apr 18 2018