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A309000 Number of strings of length n from a 3 symbol alphabet (A,B,C, say) containing at least one "A" and at least two "B"s. 0
3, 22, 105, 416, 1491, 5034, 16365, 51892, 161799, 498686, 1524705, 4635528, 14037627, 42391378, 127763925, 384536924, 1156232175, 3474201510, 10434138825, 31326533680, 94029932643, 282194655482, 846802070205, 2540859195396, 7623517110231, 22872497487694 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

This sequence can be thought of as the number of ways of rolling n 3-sided dice (with sides "A", "B", and "C") and obtaining at least one A and at least two B's.

The general formula is readily proved true by counting arguments.

LINKS

Table of n, a(n) for n=3..28.

Index entries for linear recurrences with constant coefficients, signature (9,-31,51,-40,12).

FORMULA

a(n) = 3^n - 2^(n+1) - n*2^(n-1) + n + 1.

G.f.: x^3*(-3 + 5*x)/((-1 + 3*x)*(1 - 3*x + 2*x^2)^2). - Michael De Vlieger, Jul 04 2019.

a(n) = 9*a(n-1) - 31*a(n-2) + 51*a(n-3) - 40*a(n-4) + 12*a(n-5) for n > 7. - Stefano Spezia, Jul 05 2019

EXAMPLE

Suppose three-sided dice each have sides labelled A,B,C.

If there are three dice, then ABB, BAB, and BBA are the three strings resulting from rolling the dice satisfying the property of at least one A and at least two B's, hence a(3)=3 [Note a(0)=a(1)=a(2)=0].

If there are four such dice, there are 22 such permutations, hence a(4)=22: AABB, ABAB, ABBA, ABBB, ABBC, ABCB, ACBB, BAAB, BABA, BABB, BABC, BACB, BBAA, BBAB, BBAC, BBBA, BBCA, BCAB, BCBA, CABB, CBAB, CBBA.

MATHEMATICA

Array[3^# - 2^(# + 1) - # 2^(# - 1) + # + 1 &, 27, 3] (* or *)

CoefficientList[Series[(-3 + 5 x)/((-1 + 3 x) (1 - 3 x + 2 x^2)^2), {x, 0, 26}], x] (* Michael De Vlieger, Jul 04 2019 *)

PROG

(Python) [3**n-2**(n+1)-n*2**(n-1)+n+1 for n in range(3, 20)]

(MAGMA) [3^n-2^(n+1)-n*2^(n-1)+n+1: n in [3..40]]; // Vincenzo Librandi, Jul 05 2019

CROSSREFS

Cf. A269914, A269915, A269916, A269917, A186244, A186314.

Sequence in context: A009029 A009032 A221543 * A061182 A143552 A006283

Adjacent sequences:  A308997 A308998 A308999 * A309001 A309002 A309003

KEYWORD

nonn,easy

AUTHOR

Adam Vellender, Jul 04 2019

STATUS

approved

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Last modified September 21 12:13 EDT 2020. Contains 337271 sequences. (Running on oeis4.)