A303046 Number of minimum total dominating sets in the n-Moebius ladder.
1, 6, 9, 8, 25, 3, 196, 56, 9, 20, 121, 3, 1521, 154, 9, 32, 289, 3, 5776, 300, 9, 44, 529, 3, 15625, 494, 9, 56, 841, 3, 34596, 736, 9, 68, 1225, 3, 67081, 1026, 9, 80, 1681, 3, 118336, 1364, 9, 92, 2209, 3, 194481, 1750, 9, 104, 2809, 3, 302500, 2184, 9
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
- Eric Weisstein's World of Mathematics, Moebius Ladder.
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,5,0,0,0,0,0,-10,0,0,0,0,0,10,0,0,0,0,0,-5,0,0,0,0,0,1).
Programs
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Mathematica
Table[Piecewise[{{3, Mod[n, 6] == 0}, {(n (n + 5)/6)^2, Mod[n, 6] == 1}, {n (2 n + 5)/3, Mod[n, 6] == 2}, {9, Mod[n, 6] == 3}, {2 n, Mod[n, 6] == 4}, {n^2, Mod[n, 6] == 5}}], {n, 200}] LinearRecurrence[{0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1}, {1, 6, 9, 8, 25, 3, 196, 56, 9, 20, 121, 3, 1521, 154, 9, 32, 289, 3, 5776, 300, 9, 44, 529, 3, 15625, 494, 9, 56, 841, 3}, 200] Rest @ CoefficientList[Series[3 x^6/(1 - x^6) - 9 x^3/(-1 + x^6) + 4 x^4 (2 + x^6)/(-1 + x^6)^2 - x^5 (25 + 46 x^6 + x^12)/(-1 + x^6)^3 - 2 x^2 (3 + 19 x^6 + 2 x^12)/(-1 + x^6)^3 - x (1 + 191 x^6 + 551 x^12 + 121 x^18)/(-1 + x^6)^5, {x, 0, 200}], x] -
PARI
a(n)=my(k=n\6,r=n%6);if(r<3, if(r==0, 3, if(r==1, n^2*(k+1)^2, n*(4*k+3))), if(r==3, 9, if(r==4, 2*n, n^2))) \\ Andrew Howroyd, Apr 18 2018
Formula
From Andrew Howroyd, Apr 18 2018: (Start)
a(n) = 5*a(n-6) - 10*a(n-12) + 10*a(n-18) - 5*a(n-24) + a(n-30) for n > 30.
a(6k) = 3, a(6k+1) = (6*k+1)^2*(k+1)^2, a(6k+2) = (6*k+2)*(4*k+3), a(6k+3) = 9, a(6k+4) = (6*k+4)*2, a(6k+5) = (6*k+5)^2. (End)
a(3k) = 6 - 3*(-1)^k. - Eric W. Weisstein, Apr 19 2018
Extensions
a(1)-a(2) and terms a(14) and beyond from Andrew Howroyd, Apr 18 2018
Comments