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A254027
Table T(n,k) = 3^n - 2^k read by antidiagonals.
2
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
0,2
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.
EXAMPLE
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..
.. .. .. .. ..
MATHEMATICA
Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)
PROG
(PARI) for(i=0, 10, {
for(j=0, i, print1((3^(i-j)-2^j), ", "))
});
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.
Sequence in context: A342937 A353571 A372775 * A284346 A318651 A083529
KEYWORD
sign,tabl
AUTHOR
K. G. Stier, Jan 22 2015
STATUS
approved