OFFSET
1,1
COMMENTS
The trapezohedral graph has a geometric definition for n>=3. As abstract graph it can be generalized, and the formula gives the correct results for n=1 and n=2. The sequence has been extended accordingly. - Christian Sievers, Nov 20 2023
LINKS
Christian Sievers, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Total Dominating Set
Eric Weisstein's World of Mathematics, Trapezohedral Graph
Index entries for linear recurrences with constant coefficients, signature (12,-50,66,84,-270,63,264,-128,-72,32).
FORMULA
a(n) = (A000032(n) + 2^n - 1)^2. - Christian Sievers, Nov 20 2023
G.f.: (1-4*x)^(-1)+(1-x)^(-1)+2/(1+x)+2/(-1+2*x)+(2-3*x)/(1+(-3+x)*x)-(2*(-2+x))/(-1+x+x^2)+4*(-1+x)/(-1+2*x+4*x^2). - Eric W. Weisstein, Feb 09 2024
a(n) = 12*a(n-1) - 50*a(n-2) + 66*a(n-3) + 84*a(n-4) - 270*a(n-5) + 63*a(n-6) + 264*a(n-7) - 128*a(n-8) - 72*a(n-9) + 32*a(n-10). - Eric W. Weisstein, Feb 09 2024
MATHEMATICA
Table[(LucasL[n] + 2^n - 1)^2, {n, 20}] (* Paolo Xausa, Nov 21 2023 *)
LinearRecurrence[{12, -50, 66, 84, -270, 63, 264, -128, -72, 32}, {4, 36, 121, 484, 1764, 6561, 24336, 91204, 344569, 1313316}, 20] (* Eric W. Weisstein, Feb 09 2024 *)
CoefficientList[Series[(1/(1 - 4 x) + 1/(1 - x) + 2/(1 + x) + 2/(-1 + 2 x) + (2 - 3 x)/(1 + (-3 + x) x) - (2 (-2 + x))/(-1 + x + x^2) + (4 (-1 + x))/(-1 + 2 x + 4 x^2) - 4)/x, {x, 0, 20}], x] (* Eric W. Weisstein, Feb 09 2024 *)
PROG
(PARI) a(n)=(fibonacci(n+1)+fibonacci(n-1)+2^n-1)^2 \\ Christian Sievers, Nov 20 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 02 2023
EXTENSIONS
a(1) and a(2) prepended and terms a(15) and beyond from Christian Sievers, Nov 20 2023
STATUS
approved