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A367275
a(n) = 3^(n - 1) * (9^n - 1) / 8, n >= 1.
0
1, 30, 819, 22140, 597861, 16142490, 435847959, 11767897080, 317733227721, 8578797168150, 231627523599099, 6253943137352820, 168856464709057581, 4559124547146149010, 123096362772950806239, 3323601794869686117360, 89737248461481568215441, 2422905708460002470957070
OFFSET
1,2
COMMENTS
a(n) is the number of potential matchups in a single-elimination ternary tournament (e.g., Jeopardy!) with 3^n contestants (n rounds).
Since a(n) = 3*a(n-1) + 3^(3*n - 3), the sequence in base 3 is 1, 1010, 1010100, 1010101000, 1010101010000, ....
FORMULA
a(n) = a(n) = 3*a(n-1) + 3^(3*n - 3) with a(0)=1.
a(n) = A000244(n-1)* A002452(n).
G.f.: x/((1 - 3*x)*(1 - 27*x)). - Stefano Spezia, Dec 22 2023
EXAMPLE
In a 2-round tournament, there are three first-round matches. Each first-round match has one of three possible winners, so there are 3^3 = 27 possible final matchups. Thus a(2) = 30.
MATHEMATICA
a[n_]:=3^(n-1)*(9^n-1)/8; Array[a, 20]
CROSSREFS
Sequence in context: A096722 A261030 A160269 * A367332 A049394 A143169
KEYWORD
nonn,easy
AUTHOR
Stan Seltzer, Dec 22 2023
STATUS
approved