A081335 a(n) = (6^n + 2^n)/2.
1, 4, 20, 112, 656, 3904, 23360, 140032, 839936, 5039104, 30233600, 181399552, 1088393216, 6530351104, 39182090240, 235092508672, 1410554986496, 8463329787904, 50779978465280, 304679870267392, 1828079220555776
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (8,-12).
Crossrefs
Cf. A081336.
Programs
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GAP
List([0..30], n-> 2^(n-1)*(3^n + 1)); # G. C. Greubel, Aug 02 2019
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Magma
[(6^n+2^n)/2: n in [0..30]]; // Vincenzo Librandi, Aug 08 2013
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Mathematica
LinearRecurrence[{8, -12}, {1, 4}, 30] (* Harvey P. Dale, May 03 2013 *) CoefficientList[Series[(1-4x)/((1-2x)(1-6x)), {x,0,30}], x] (* Vincenzo Librandi, Aug 08 2013 *) -
PARI
a(n)=(6^n+2^n)/2 \\ Charles R Greathouse IV, Oct 07 2015
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Sage
[2^(n-1)*(3^n + 1) for n in (0..30)] # G. C. Greubel, Aug 02 2019
Formula
a(n) = 8*a(n-1) - 12*a(n-2), a(0)=1, a(1)=4.
G.f.: (1-4*x)/((1-2*x)*(1-6*x)).
E.g.f.: exp(4*x)*cosh(2*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * 4^(n-k) = Sum_{k=0..n} binomial(n,k) * 4^(n-k/2) * (1+(-1)^k)/2. - Paul Barry, Nov 22 2003
a(n) = Sum_{k=0..n} 4^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
Comments