A214091 -id:A214091 - 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

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Original entry on oeis.org

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

A239926 3^(p-1)-2^(p+1) for primes p > 3.

Original entry on oeis.org

17, 473, 54953, 515057, 42784577, 386371913, 31364282393, 22875718713137, 205886837127353, 150094360419092177, 12157661061010417697, 109418971539326314793, 8862937838177524385273, 6461081871212274789450257, 4710128696093323330314756713

Offset: 1

Vincenzo Librandi, Jun 17 2014

3^(p-1)-2^(p+1) can be written as (3^((p-1)/2)-2^((p+1)/2))*(3^((p-1)/2)+2^((p+1)/2)). Since 3^((p-1)/2)-2^((p+1)/2) > 1 for p > 5, these numbers are all composite after 17 = (3^2-2^3)*(3^2+2^3).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040, A003063, A135171 (numbers of the form 3^p-2^p with p prime), A214091 (supersequence).

[3^(p-1)-2^(p+1): p in PrimesInInterval(4,100)];

Table[3^(Prime[n] - 1) - 2^(Prime[n] + 1), {n, 3, 100}]

cp's OEIS Frontend

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

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A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Views

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Comments

Examples

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Crossrefs

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Mathematica

PARI

A239926 3^(p-1)-2^(p+1) for primes p > 3.

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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica