A154120 -id:A154120 - cp's OEIS frontend

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

N. J. A. Sloane, Simon Plouffe

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

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

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).

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

Cf. A000217, A000244, A027472, A116138,

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

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

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 *)

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

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)

A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).

Original entry on oeis.org

6, 96, 960, 7680, 53760, 344064, 2064384, 11796480, 64880640, 346030080, 1799356416, 9160359936, 45801799680, 225485783040, 1095216660480, 5257039970304, 24970939858944, 117510305218560, 548381424353280, 2539871860162560, 11683410556747776, 53409876830846976

Offset: 4

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
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 (16,-96,256,-256).

Cf. A000142, A006043, A038846, A154120.

Column k=3 of square array A152818. - Paul Curtz, Dec 17 2008 [corrected by Omar E. Pol, Jan 07 2009]

[4^(n-4)*(n-3)*(n-2)*(n-1): n in [4..30]]; // Vincenzo Librandi, Aug 14 2011

a[n_] := 4^(n - 4)*(n - 1)*(n - 2)*(n - 3); Array[a, 25, 4] (* Amiram Eldar, Jan 08 2023 *)

G.f. = 6*x^4/(1-4*x)^4. - Emeric Deutsch, Apr 29 2004
a(n) = 6*A038846(n). - R. J. Mathar , Mar 22 2013
E.g.f.: (3 + exp(4*x)*(32*x^3 - 24*x^2 + 12*x - 3))/128. - Stefano Spezia, Jan 01 2023
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=4} 1/a(n) = 18*log(4/3) - 5.
Sum_{n>=4} (-1)^n/a(n) = 50*log(5/4) - 11. (End)

More terms from Emeric Deutsch, Apr 29 2004
Erroneous reference deleted by Martin J. Erickson (erickson(AT)truman.edu), Nov 03 2010
Entry revised by N. J. A. Sloane, Dec 27 2021

A154128 a(n) = 5^n*(n+4)!/n!.

Original entry on oeis.org

24, 600, 9000, 105000, 1050000, 9450000, 78750000, 618750000, 4640625000, 33515625000, 234609375000, 1599609375000, 10664062500000, 69726562500000, 448242187500000, 2838867187500000, 17742919921875000, 109588623046875000

Offset: 0

Omar E. Pol, Jan 05 2009

Column 4 of square array A152818.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (25, -250, 1250, -3125, 3125).

Cf. A000142, A006043, A006044, A152818, A154120.

[5^n*(n+4)*(n+3)*(n+2)*(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

LinearRecurrence[{25, -250, 1250, -3125, 3125}, {24, 600, 9000, 105000, 1050000}, 25] (* or *) Table[5^n*(n+4)*(n+3)*(n+2)*(n+1), {n,0,25}] (* G. C. Greubel, Sep 02 2016 *)

a(n) = 5^n*(n+4)*(n+3)*(n+2)*(n+1).
From R. J. Mathar, Feb 06 2009: (Start)
a(n) = A052762(n+4)*A000351(n).
a(n) = 24*A036071(n).
G.f: 24/(1-5*x)^5. (End)
From G. C. Greubel, Sep 02 2016: (Start)
a(n) = 25*a(n-1) - 250*a(n-2) + 1250*a(n-3) - 3125*a(n-4) + 3125*a(n-5).
E.g.f.: (24 + 480*x + 1800*x^2 + 2000*x^3 + 625*x^4)*exp(5*x). (End)

More terms from R. J. Mathar, Feb 06 2009

cp's OEIS Frontend

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

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A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).

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A154128 a(n) = 5^n*(n+4)!/n!.

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cp's OEIS Frontend

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

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Magma

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A006044 a(n) = 4^(n-4)*(n-1)*(n-2)*(n-3).

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Magma

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A154128 a(n) = 5^n*(n+4)!/n!.

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Mathematica

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Extensions

A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).