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A362751
Number of total dominating sets in the n-trapezohedral graph.
1
4, 36, 121, 484, 1764, 6561, 24336, 91204, 344569, 1313316, 5044516, 19509889, 75898944, 296735076, 1164925161, 4588978564, 18128468164, 71781662241, 284767380496, 1131461944804, 4501301127129, 17925960016836, 71447900614596, 284964683524609, 1137186233284864
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
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
Cf. A000032.
Sequence in context: A193874 A254939 A038688 * A076830 A144298 A072109
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