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

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. A000217, A000244, A027472, A116138,

Programs

Magma

[(n+2)*(n+1)*3^n: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

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 *)

PARI

a(n)=(n+2)*(n+1)*3^n \\ Charles R Greathouse IV, Mar 16 2011

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)