OFFSET
1,2
LINKS
Christian Sievers, Table of n, a(n) for n = 3..1000
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Eric Weisstein's World of Mathematics, Trapezohedral Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,-1,14,-6,-5,-3,7,-4,3,0,0,-2,1).
FORMULA
a(n) = (A001608(n) + n)^2. - Christian Sievers, Jun 14 2024
From Eric W. Weisstein, Dec 12 2024: (Start)
a(n) = 4*a(n-1)-3*a(n-2)-4*a(n-3)-a(n-4)+14*a(n-5)-6*a(n-6)-5*a(n-7)-3*a(n-8)+7*a(n-9)-4*a(n-10)+3*a(n-11)-2*a(n-14)+a(n-15).
G.f.: -(x*(1+12*x-25*x^2-56*x^3+129*x^4-25*x^5-26*x^6-64*x^7+112*x^8-121*x^9+74*x^10-x^11-7*x^12-14*x^13+9*x^14)/((-1+x)^3*(-1-x+x^3)*(-1+2*x-x^2+x^3)*(-1+x^2+x^3)^2)). (End)
MATHEMATICA
Table[(RootSum[-1 - # + #^3 &, #^n &] + n)^2, {n, 20}] (* Eric W. Weisstein, Dec 12 2024 *)
LinearRecurrence[{4, -3, -4, -1, 14, -6, -5, -3, 7, -4, 3, 0, 0, -2, 1}, {1, 16, 36, 36, 100, 121, 196, 324, 441, 729, 1089, 1681, 2704, 4225, 6889}, 20] (* Eric W. Weisstein, Dec 12 2024 *)
CoefficientList[Series[-(1 + 12 x - 25 x^2 - 56 x^3 + 129 x^4 - 25 x^5 - 26 x^6 - 64 x^7 + 112 x^8 - 121 x^9 + 74 x^10 - x^11 - 7 x^12 - 14 x^13 + 9 x^14)/((-1 + x)^3 (-1 - x + x^3) (-1 + 2 x - x^2 + x^3) (-1 + x^2 + x^3)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 12 2024 *)
PROG
(PARI) a(n)=(polsym(x^3-x-1, n)[n+1]+n)^2 \\ Christian Sievers, Jun 14 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, May 11 2024
EXTENSIONS
a(13) and beyond from Christian Sievers, Jun 14 2024
a(1)-a(2) prepended by Eric W. Weisstein, Dec 12 2024
Offset corrected for above change
STATUS
approved