OFFSET
1,4
COMMENTS
Binomial transform of A000918: (-1, 0, 2, 6, 14, 30, ...). - Gary W. Adamson, Mar 23 2012
This sequence demonstrates 2^n as a loose lower bound for g(n) in Waring's problem. Since 3^n > 2(2^n) for all n > 2, the number 2^(n + 1) - 1 requires 2^n n-th powers for its representation since 3^n is not available for use in the sum: the gulf between the relevant powers of 2 and 3 widens considerably as n gets progressively larger. - Alonso del Arte, Feb 01 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. Knuth, Letter to N. J. A. Sloane, date unknown
Index entries for linear recurrences with constant coefficients, signature (5,-6).
FORMULA
Let b(n) = 2*(3/2)^n - 1. Then a(n) = -b(1-n)*3^(n-1) for n > 0. A083313(n) = A064686(n) = b(n)*2^(n-1) for n > 0. - Michael Somos, Aug 06 2006
From Colin Barker, May 27 2013: (Start)
a(n) = 5*a(n-1) - 6*a(n-2).
G.f.: -x*(1-4*x) / ((1-2*x)*(1-3*x)). (End)
E.g.f.: (1/3)*(2 - 3*exp(2*x) + exp(3*x)). - G. C. Greubel, Nov 03 2022
EXAMPLE
a(3) = 1 because 3^2 - 2^3 = 9 - 8 = 1.
a(4) = 11 because 3^3 - 2^4 = 27 - 16 = 11.
a(5) = 49 because 3^4 - 2^5 = 81 - 32 = 49.
MATHEMATICA
Table[3^(n-1) - 2^n, {n, 25}] (* Alonso del Arte, Feb 01 2013 *)
LinearRecurrence[{5, -6}, {-1, -1}, 30] (* Harvey P. Dale, Feb 02 2015 *)
PROG
(PARI) a(n)=3^(n-1)-2^n \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [3^(n-1) -2^n: n in [1..30]]; // G. C. Greubel, Nov 03 2022
(SageMath) [3^(n-1) -2^n for n in range(1, 31)] # G. C. Greubel, Nov 03 2022
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Henrik Johansson (Henrik.Johansson(AT)Nexus.SE)
EXTENSIONS
A few more terms from Alonso del Arte, Feb 01 2013
STATUS
approved