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A245023 Number of cases of tie (no winner) in the n-person rock-paper-scissors game. 1
3, 3, 9, 39, 153, 543, 1809, 5799, 18153, 55983, 171009, 519159, 1569753, 4733823, 14250609, 42850119, 128746953, 386634063, 1160688609, 3483638679, 10454061753, 31368476703, 94118013009, 282379204839, 847187946153, 2541664501743, 7625194831809, 22875987148599, 68628766752153, 205887910869183, 617666953833009 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..31.

Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).

FORMULA

a(n) = 3^n - 3*(2^n-2) = 3 * A101052(n-1), n >= 1.

a(n) = 5*a(n-1)-5*a(n-2)-5*a(n-3)+6*a(n-4). - Colin Barker, Jul 26 2014

G.f.: -3*x*(8*x^2-5*x+1) / ((x-1)*(2*x-1)*(3*x-1)). - Colin Barker, Jul 26 2014

a(n) = 3 + 3!*S2(n, 3) with S2(n, k) = A008277(n, k) (Stirling numbers of the second kind). S2(n,3)= A000392(n). Proof: Use the formula given in A000392. Hint for an independent proof:  consider the partition array A036040 for the multinomial M_3 numbers. Only partitions of n with number of parts m = 1, 2 and 3 matter here. Each partition defines a pattern for a multilist, like 2^1,3^2 defines the n=8, m=3 pattern [..][...][...]. The corresponding M_3 number 280 = C(8,2)*C(6,3)/2 gives the number of possibilities to form from objects, here R,R,P,P,P,S,S,S, lists of length 8 (the order is relevant).  If m=1 then M_3 = 1 and for n one has 3 lists [n times R], [n times P] and [n times S] (no winner), If m=2 or 3 each of the M_3(n,m,j) (j=1..p(n,m), the number of partitions of n with m parts) comes 3! times from the permutation of the R, P and S symbols. The sum of the M_3 numbers over like m gives the Stirling2 numbers. If m=2 there are always winners for each n (only two symbols are present in each list). If m=1 or m=3 there is no winner.  - Wolfdieter Lang, Aug 01 2014

EXAMPLE

R, P, S = each Rock, Paper, Scissors. For n = 2 RR, PP, SS. 3 cases. and for n = 3 RRR, PPP, SSS, RPS, PRS, RSP, PSR, SPR, SRP. 9 cases.

RRS (and RSR, SRR) is not a tie case because there are two winners. SPP (and PPS, PSP) is not a tie case because there is a winner even though the 2nd and 3rd places cannot be determined.  - Wolfdieter Lang, Jul 31 2014

MAPLE

A245023:=n->3^n - 3*(2^n-2): seq(A245023(n), n=1..30); # Wesley Ivan Hurt, Jul 26 2014

MATHEMATICA

Table[3^n - 3 (2^n - 2), {n, 30}] (* Wesley Ivan Hurt, Jul 26 2014 *)

PROG

(PARI) a(n) = 3^n - 3*(2^n-2); \\ Michel Marcus, Jul 16 2014

(MAGMA) [3^n - 3*(2^n-2) : n in [1..30]]; // Wesley Ivan Hurt, Jul 26 2014

CROSSREFS

Cf. A101052.

Sequence in context: A032086 A241278 A100239 * A038080 A257621 A190960

Adjacent sequences:  A245020 A245021 A245022 * A245024 A245025 A245026

KEYWORD

nonn,easy

AUTHOR

Jaeyool Park, Jul 10 2014

EXTENSIONS

Typo in data fixed by Colin Barker, Jul 26 2014

STATUS

approved

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Last modified May 25 06:50 EDT 2020. Contains 334584 sequences. (Running on oeis4.)