OFFSET
1,1
COMMENTS
The n-double cone graph is defined for n >= 3. The sequence has been extended to n=1 using the formula/recurrence. - Andrew Howroyd, May 03 2023
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Double Cone Graph
Eric Weisstein's World of Mathematics, Total Dominating Set
Index entries for linear recurrences with constant coefficients, signature (7,-15,18,-24,-6,27,-15,13,-4).
FORMULA
From Andrew Howroyd, May 03 2023: (Start)
a(n) = A001638(n)^2 + 4^n - 1.
a(2*n-1) = A302603(4*n-1).
a(n) = 7*a(n-1) - 15*a(n-2) + 18*a(n-3) - 24*a(n-4) - 6*a(n-5) + 27*a(n-6) - 15*a(n-7) + 13*a(n-8) - 4*a(n-9) for n > 9.
G.f.: x*(4 - 12*x + 27*x^2 - 49*x^3 - 215*x^4 + 369*x^5 - 237*x^6 + 207*x^7 - 64*x^8)/((1 - x)*(1 + x)*(1 - 4*x)*(1 - 3*x + x^2)*(1 + 3*x^2 + x^4)).
(End)
MATHEMATICA
Table[1 + 4 (-1)^n + 4^n + LucasL[2 n] + 4 LucasL[n] Cos[n Pi/2], {n, 20}] (* Eric W. Weisstein, Sep 09 2023 *)
Table[(2 Cos[n Pi/2] + Fibonacci[n + 1] + Fibonacci[n - 1])^2 + 4^n - 1, {n, 20}] (* Eric W. Weisstein, Sep 09 2023 *)
LinearRecurrence[{7, -15, 18, -24, -6, 27, -15, 13, -4}, {4, 16, 79, 336, 1144, 4351, 17224, 67936, 267919}, 20] (* Eric W. Weisstein, Sep 09 2023 *)
CoefficientList[Series[4/(1 - 4 x) + 1/(1 - x) - 4/(1 + x) + (3 - 2 x)/(1 + (-3 + x) x) - 4 x (3 + 2 x^2)/(1 + 3 x^2 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 09 2023 *)
PROG
(PARI) a(n) = {(fibonacci(n+1) + fibonacci(n-1) + I^n + (-I)^n)^2 + 4^n - 1} \\ Andrew Howroyd, May 03 2023
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Eric W. Weisstein, May 02 2023
EXTENSIONS
a(1)-a(2) prepended and a(16) and beyond from Andrew Howroyd, May 03 2023
STATUS
approved