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

-1, -1, 1, 11, 49, 179, 601, 1931, 6049, 18659, 57001, 173051, 523249, 1577939, 4750201, 14283371, 42915649, 128878019, 386896201, 1161212891, 3484687249, 10456158899, 31372671001, 94126401611, 282395982049, 847221500579, 2541731610601, 7625329049531, 22876255584049

Offset: 1

Author

Henrik Johansson (Henrik.Johansson(AT)Nexus.SE)

Keywords

• sign
• easy

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

Examples

			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.
		

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).

Crossrefs

Cf. A000918, A056182 (first differences), A064686, A083313, A214091, A369490.

Cf. A363024 (prime terms).

From the third term onward the first differences of A005173.

Difference between two leftmost columns of A090888.

A diagonal in A254027.

Right edge of irregular triangle A252750.

Programs

Magma

[3^(n-1) -2^n: n in [1..30]]; // G. C. Greubel, Nov 03 2022

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 *)

PARI

a(n)=3^(n-1)-2^n \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[3^(n-1) -2^n for n in range(1,31)] # G. C. Greubel, Nov 03 2022

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

Extensions

A few more terms from Alonso del Arte, Feb 01 2013