A220087 -id:A220087 - cp's OEIS frontend

A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

21, 37, 53, 69, 101, 117, 133, 197, 229, 245, 261, 389, 453, 485, 501, 517, 773, 901, 965, 997, 1013, 1029, 1541, 1797, 1925, 1989, 2021, 2037, 2053, 3077, 3589, 3845, 3973, 4037, 4069, 4085, 4101, 6149, 7173, 7685, 7941, 8069, 8133, 8165, 8181, 8197, 12293, 14341, 15365, 15877, 16133

Offset: 1

Brad Clardy, Apr 16 2013

The table has row labels 2^n - 1 and column labels 2^(m+3). The table entry is row*col + 5. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+5  |   16    32    64   128   256    512   1024 ...
----|-------------------------------------------
1   |   21    37    69   133   261    517   1029
3   |   53   101   197   389   773   1541   3077
7   |  117   229   453   901  1797   3589   7173
15  |  245   485   965  1925  3845   7685  15365
31  |  501   997  1989  3973  7941  15877  31749
63  | 1013  2021  4037  8069 16133  32261  64517
127 | 2037  4069  8133 16261 32517  65029 130053
...
All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is a repeating series
  of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m. The total number of 1's and 5's in the pattern is 2^(j+2) - 1, where j is the column index.
As an example, consider A(1), which is 21; the sequence B(n) where i XOR 20 = i - 20 starts as 20, 21, 22, 23, 28, 29, 30, 31, 52, ... with successive differences of 1, 1, 1, 5, 1, 1, 1, 21.
for A(2), which is 37, the sequence B(n) where i XOR 36 = i - 36 starts as 36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 100, ... with successive differences of 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 37.

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A057555 (lexicographic ordering).
Rows: A168614(i=1), n>=4.
Cols: A220087(j=2), n>=6.

//program generates values in a table form, row labels of 2^i -1
for i:=1 to 10 do
    m:=2^i - 1;
    m, [ m*2^(n+3) +5 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(3+i-j) + 5;
       if IsPrime(k) then k, "*";
          else k;
       end if;
    end for;
end for;

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 3) + 5 for n>=1.

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),", "))
});

A220088 a(n) = 2^n - 81.

Original entry on oeis.org

-80, -79, -77, -73, -65, -49, -17, 47, 175, 431, 943, 1967, 4015, 8111, 16303, 32687, 65455, 130991, 262063, 524207, 1048495, 2097071, 4194223, 8388527, 16777135, 33554351, 67108783, 134217647, 268435375, 536870831, 1073741743, 2147483567, 4294967215

Offset: 0

Andreas Rieber, Dec 04 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A185346, A220087.

Table[2^n - 81, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (161*x - 80)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 11 2023: (Start)
a(n) = 2*a(n-1) + 81 with a(0) = -80.
E.g.f.: exp(2*x) - 81*exp(x). (End)

A220089 a(n) = 2^n - 243.

Original entry on oeis.org

-242, -241, -239, -235, -227, -211, -179, -115, 13, 269, 781, 1805, 3853, 7949, 16141, 32525, 65293, 130829, 261901, 524045, 1048333, 2096909, 4194061, 8388365, 16776973, 33554189, 67108621, 134217485, 268435213, 536870669, 1073741581, 2147483405, 4294967053

Offset: 0

Andreas Rieber, Dec 04 2012

sign

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A185346, A220087, A220088.

Table[2^n - 243, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (485*x - 242)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 11 2023: (Start)
a(n) = 2*a(n-1) + 243 with a(0) = -242.
E.g.f.: exp(2*x) - 243*exp(x). (End)

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A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

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

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A220088 a(n) = 2^n - 81.

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A220089 a(n) = 2^n - 243.

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