OFFSET
1,1
COMMENTS
The n-trapezohedral graph is defined for n >= 3. The sequence has been extended to a(1) using the formula. - Andrew Howroyd, Mar 20 2025
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Trapezohedral Graph.
Index entries for linear recurrences with constant coefficients, signature (12,-52,88,2,-176,161,10,-60,16).
FORMULA
From Andrew Howroyd, Mar 20 2025: (Start)
G.f.: x*(8 - 60*x + 99*x^2 + 224*x^3 - 832*x^4 + 707*x^5 - 10*x^6 - 76*x^7)/((1 - x)^2*(1 - 4*x)*(1 - 3*x + x^2)*(1 - x - x^2)*(1 - 2*x - 4*x^2)). (End)
a(n) = 4^n + 4*n + 1 + 2*(2^n - 1)*Lucas(n) - Lucas(2*n). - Eric W. Weisstein, Sep 03 2025
a(n) = 12*a(n-1)-52*a(n-2)+88*a(n-3)+2*a(n-4)-176*a(n-5)+161*a(n-6)+10*a(n-7)-60*a(n-8)+16*a(n-9). - Eric W. Weisstein, Sep 03 2025
MATHEMATICA
Table[4^n + 4 n + 1 + 2 (2^n - 1) LucasL[n] - LucasL[2 n], {n, 20}] (* Eric W. Weisstein, Sep 03 2025 *)
LinearRecurrence[{12, -52, 88, 2, -176, 161, 10, -60, 16}, {8, 36, 115, 436, 1604, 6067, 22936, 87332, 334075}, 20] (* Eric W. Weisstein, Sep 03 2025 *)
CoefficientList[Series[(-8 + 60 x - 99 x^2 - 224 x^3 + 832 x^4 - 707 x^5 + 10 x^6 + 76 x^7)/((-1 + x)^2 (-1 + 4 x) (1 - 3 x + x^2) (-1 + x + x^2) (-1 + 2 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 03 2025 *)
PROG
(PARI) \\ here b(n) = A000032(n).
b(n) = fibonacci(n+1) + fibonacci(n-1)
a(n) = 4^n - b(2*n) + 2*(2^n-1)*b(n) + 1 + 4*n \\ Andrew Howroyd, Mar 20 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Mar 07 2025
EXTENSIONS
a(1)-a(2) prepended and a(14) onwards from Andrew Howroyd, Mar 20 2025
STATUS
approved