A006043 A traffic light problem: expansion of 2/(1 - 3*x)^3. (Formerly M2107)

2, 18, 108, 540, 2430, 10206, 40824, 157464, 590490, 2165130, 7794468, 27634932, 96722262, 334807830, 1147912560, 3902902704, 13172296626, 44165935746, 147219785820, 488149816140, 1610894393262, 5292938720718, 17322344904168, 56485907296200, 183579198712650, 594796603828986

Offset

0,1

Comments

Column 2 of square array A152818. - Omar E. Pol, Jan 05 2009

In [Bach et al., Section 9], 2*a(n-2) counts the "small diagrams". - Eric M. Schmidt, Sep 23 2017

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Eric Bach, Jeremie Dusart, Lisa Hellerstein, and Devorah Kletenik, Submodular Goal Value of Boolean Functions, arXiv:1702.04067 [cs.DM], 2017.

Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.

Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)

Frank A. Haight, Letter to N. J. A. Sloane, n.d..

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Formula

a(n) = (n+2)*(n+1)*3^n. - Zerinvary Lajos, Apr 25 2007, corrected by R. J. Mathar, Mar 14 2011

a(n) = 2*A027472(n+3) = A116138(n+1)/3. - R. J. Mathar, Mar 14 2011

a(n) = 2*A000217(n+1)*A000244(n). - Zak Seidov, Mar 14 2011

E.g.f.: exp(3*x)*(2 + 12*x + 9*x^2). - Stefano Spezia, Jan 01 2023

From Amiram Eldar, Jan 08 2023: (Start)

Sum_{n>=0} 1/a(n) = 3 - 6*log(3/2).

Sum_{n>=0} (-1)^n/a(n) = 12*log(4/3) - 3. (End)

Maple

seq((n+2)*(n+1)*3^n, n=0..23); # Zerinvary Lajos, Apr 25 2007

Mathematica

f[n_] := (n + 2) (n + 1) 3^n; Array[f, 22, 0] (* Robert G. Wilson v, Mar 15 2011 *)
CoefficientList[Series[2/(1 - 3 x)^3, {x, 0, 21}], x] (* Robert G. Wilson v, Mar 15 2011 *)
LinearRecurrence[{9, -27, 27}, {2, 18, 108}, 30] (* Harvey P. Dale, Apr 27 2017 *)

Prog

(PARI) a(n)=(n+2)*(n+1)*3^n \\ Charles R Greathouse IV, Mar 16 2011
(Magma)[(n+2)*(n+1)*3^n: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

Crossrefs

Cf. A006044, A000142, A152818, A154120. - Omar E. Pol, Jan 05 2009

Cf. A000217, A000244, A027472, A116138,

Sequence in context: A094251 A345969 A101570 * A112328 A370732 A038721

Adjacent sequences: A006040 A006041 A006042 * A006044 A006045 A006046

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved