A116144 a(n) = 4^n * n*(n+1).
0, 8, 96, 768, 5120, 30720, 172032, 917504, 4718592, 23592960, 115343360, 553648128, 2617245696, 12213813248, 56371445760, 257698037760, 1168231104512, 5257039970304, 23502061043712, 104453604638720, 461794883665920
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (12,-48,64).
Programs
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GAP
List([0..30], n-> 4^n*n*(n+1)); # G. C. Greubel, May 10 2019
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Magma
[(n^2+n)*4^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013
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Magma
I:=[0,8,96]; [n le 3 select I[n] else 12*Self(n-1)-48*Self(n-2)+64*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 28 2013
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Mathematica
Table[(n^2 + n)*4^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 28 2013 *) LinearRecurrence[{12,-48,64},{0,8,96},30] (* Harvey P. Dale, Feb 27 2015 *) -
PARI
a(n)=(n^2+n)*4^n \\ Charles R Greathouse IV, Feb 28 2013
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Sage
[4^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019
Formula
From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 8*x/(1-4*x)^3.
a(n) = 8*A038845(n-1). (End)
a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3). - Vincenzo Librandi, Feb 28 2013
E.g.f.: 8*x*(1 + 2*x)*exp(4*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 3*log(4/3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5/4) - 1. (End)