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

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

Author

K. G. Stier, Jan 22 2015

Keywords

• sign
• tabl

Comments

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.

Examples

			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..
  ..   ..  ..  ..  ..
		

Links

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

Crossrefs

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.

Programs

Mathematica

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

PARI

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