A254027 - OEIS

%I #26 Dec 18 2017 19:46:40

%S 0,2,-1,8,1,-3,26,7,-1,-7,80,25,5,-5,-15,242,79,23,1,-13,-31,728,241,

%T 77,19,-7,-29,-63,2186,727,239,73,11,-23,-61,-127,6560,2185,725,235,

%U 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

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

%C 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.

%C 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.

%H K. G. Stier, <a href="/A254027/b254027.txt">Table of n, a(n) for n = 0..5150</a>

%e Table begins

%e    0    2   8  26  80..

%e   -1    1   7  25  79..

%e   -3   -1   5  23  73..

%e   -7   -5   1  19  65..

%e   -15 -13  -7  11  49..

%e   ..   ..  ..  ..  ..

%t Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 18 2017 *)

%o (PARI) for(i=0, 10, {

%o      for(j=0, i, print1((3^(i-j)-2^j),", "))

%o });

%Y Row 0 (=3^n-1) is A024023.

%Y Row 1 (=3^n-2) is A058481.

%Y Row 2 (=3^n-4) is A168611.

%Y Column 0 (=1-2^n) is (-1)A000225.

%Y Column 1 (=3-2^n) is (-1)A036563.

%Y Column 2 (=9-2^n) is (-1)A185346.

%Y Column 3 (=27-2^n) is (-1)A220087.

%Y 0,0-Diagonal (=3^n-2^n) is A001047.

%Y 1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.

%Y 0,1-Diagonal (=3^n-2^(n+1)) is A003063.

%Y 0,2-Diagonal (=3^n-2^(n+2)) is A214091.

%K sign,tabl

%O 0,2

%A _K. G. Stier_, Jan 22 2015