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%I #42 Nov 04 2022 07:30:59
%S -1,-1,1,11,49,179,601,1931,6049,18659,57001,173051,523249,1577939,
%T 4750201,14283371,42915649,128878019,386896201,1161212891,3484687249,
%U 10456158899,31372671001,94126401611,282395982049,847221500579,2541731610601,7625329049531,22876255584049
%N a(n) = 3^(n-1) - 2^n.
%C Binomial transform of A000918: (-1, 0, 2, 6, 14, 30, ...). - _Gary W. Adamson_, Mar 23 2012
%C 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
%H Vincenzo Librandi, <a href="/A003063/b003063.txt">Table of n, a(n) for n = 1..1000</a>
%H D. Knuth, <a href="/A003063/a003063.pdf">Letter to N. J. A. Sloane, date unknown</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F 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
%F From _Colin Barker_, May 27 2013: (Start)
%F a(n) = 5*a(n-1) - 6*a(n-2).
%F G.f.: -x*(1-4*x) / ((1-2*x)*(1-3*x)). (End)
%F E.g.f.: (1/3)*(2 - 3*exp(2*x) + exp(3*x)). - _G. C. Greubel_, Nov 03 2022
%e a(3) = 1 because 3^2 - 2^3 = 9 - 8 = 1.
%e a(4) = 11 because 3^3 - 2^4 = 27 - 16 = 11.
%e a(5) = 49 because 3^4 - 2^5 = 81 - 32 = 49.
%t Table[3^(n-1) - 2^n, {n, 25}] (* _Alonso del Arte_, Feb 01 2013 *)
%t LinearRecurrence[{5,-6},{-1,-1},30] (* _Harvey P. Dale_, Feb 02 2015 *)
%o (PARI) a(n)=3^(n-1)-2^n \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [3^(n-1) -2^n: n in [1..30]]; // _G. C. Greubel_, Nov 03 2022
%o (SageMath) [3^(n-1) -2^n for n in range(1,31)] # _G. C. Greubel_, Nov 03 2022
%Y Cf. A000918, A064686, A083313.
%K sign,easy
%O 1,4
%A Henrik Johansson (Henrik.Johansson(AT)Nexus.SE)
%E A few more terms from _Alonso del Arte_, Feb 01 2013
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