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A003063 a(n) = 3^(n-1)-2^n. 14

%I

%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 a(n) = 5*a(n-1)-6*a(n-2). G.f.: x*(4*x-1) / ((2*x-1)*(3*x-1)). - _Colin Barker_, May 27 2013

%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

%Y Cf. A000918.

%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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Last modified June 14 13:23 EDT 2021. Contains 345025 sequences. (Running on oeis4.)