A363024 -id:A363024 - cp's OEIS frontend

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

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

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

			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.

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).

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.

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

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 *)

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

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

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

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

A363375 Numbers k such that 3^(k-1) - 2^k is prime.

Original entry on oeis.org

4, 6, 7, 8, 22, 32, 45, 52, 58, 60, 85, 98, 211, 290, 291, 426, 428, 712, 903, 1392, 1683, 1828, 2342, 3482, 4818, 4887, 9060, 14328, 16948, 17581, 18358, 65298, 69237, 84770, 94788

Offset: 1

Sébastien Tao, May 29 2023

a(36) > 100000. - Hugo Pfoertner, Jun 03 2023

			a(1) = 4 is in the sequence because 3^3 - 2^4 =  11 is prime.
a(2) = 6 is in the sequence because 3^5 - 2^6 = 179 is prime.

Cf. A057468, A162714, A363024.

The sequence that results from increasing all terms by 1 in A162714 is a subsequence.

Cases[Range[1, 300], k_ /; PrimeQ[3^(k - 1) - 2^k]]

a(16)-a(31) from Michael S. Branicky, May 29 2023
a(32)-a(33) from Hugo Pfoertner, May 29 2023
a(34)-a(35) from Hugo Pfoertner, Jun 02 2023

cp's OEIS Frontend

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

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