A020515 -id:A020515 - cp's OEIS frontend

A155599 a(n) = 8^n - 2^n + 1^n.

1, 7, 61, 505, 4081, 32737, 262081, 2097025, 16776961, 134217217, 1073740801, 8589932545, 68719472641, 549755805697, 4398046494721, 35184372056065, 281474976645121, 2251799813554177, 18014398509219841, 144115188075331585

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (11,-26,16).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597, A155598.

Table[8^n-2^n+1,{n,0,30}] (* or *) LinearRecurrence[{11,-26,16},{1,7,61},30] (* Harvey P. Dale, Feb 25 2014 *)

a(n)=8^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-8*x) - 1/(1-2*x) + 1/(1-x).
E.g.f.: e^(8*x) - e^(2*x) + e^x.
a(n) = 10*a(n-1) - 16*a(n-2) + 7 with a(0)=1, a(1)=7 - Vincenzo Librandi, Jul 21 2010
a(n) = A248217(n)+1. - R. J. Mathar, Mar 10 2022

A155596 a(n) = 5^n - 2^n + 1^n.

Original entry on oeis.org

1, 4, 22, 118, 610, 3094, 15562, 77998, 390370, 1952614, 9764602, 48826078, 244136530, 1220694934, 6103499242, 30517545358, 152587825090, 762939322054, 3814697003482, 19073485803838, 95367430592050, 476837156105974

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594.

Table[5^n-2^n+1,{n,0,30}] (* or *) LinearRecurrence[{8,-17,10},{1,4,22},30] (* Harvey P. Dale, Sep 11 2019 *)

a(n)=5^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-5*x)-1/(1-2*x)+1/(1-x).
E.g.f.: e^(5*x) - e^(2*x) + e^x.
a(n) = 7*a(n-1)-10*a(n-2)+4 with a(0)=1, a(1)=4. - Vincenzo Librandi, Jul 21 2010
a(n) = A005057(n)+1. - R. J. Mathar, Mar 10 2022

A155597 a(n) = 6^n - 2^n + 1.

Original entry on oeis.org

1, 5, 33, 209, 1281, 7745, 46593, 279809, 1679361, 10077185, 60465153, 362795009, 2176778241, 13060685825, 78364147713, 470184951809, 2821109841921, 16926659313665, 101559956406273, 609359739486209, 3656158439014401, 21936950638280705, 131621703838072833

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596.

Table[6^n-2^n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-20,12},{1,5,33},20] (* Harvey P. Dale, Jul 13 2011 *)

a(n) = 6^n-2^n+1 \\ Charles R Greathouse IV, Jul 25 2011

G.f.: 1/(1-6*x)-1/(1-2*x)+1/(1-x).
E.g.f.: exp(6*x)-exp(2*x)+exp(x).
a(n) = 8*a(n-1)-12*a(n-2)+5 with a(0)=1, a(1)=5. - Vincenzo Librandi, Jul 21 2010
a(0)=1, a(1)=5, a(2)=33, a(n) = 9*a(n-1)-20*a(n-2)+12*a(n-3). - Harvey P. Dale, Jul 13 2011
a(n) = A248216(n)+1. - R. J. Mathar, Mar 10 2022

A155600 a(n) = 9^n-2^n+1^n.

Original entry on oeis.org

1, 8, 78, 722, 6546, 59018, 531378, 4782842, 43046466, 387419978, 3486783378, 31381057562, 282429532386, 2541865820138, 22876792438578, 205891132061882, 1853020188786306, 16677181699535498, 150094635296736978

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-29,18).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597, A155598, A155599.

Table[9^n - 2^n + 1, {n, 0, 25}] (* or *)
LinearRecurrence[{12, -29, 18}, {1, 8, 78}, 26] (* Paolo Xausa, Jul 19 2024 *)

a(n)=9^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-9*x)-1/(1-2*x)+1/(1-x). E.g.f.: e^(9*x)-e^(2*x)+e^x.
a(n) = 11*a(n-1)-18*a(n-2)+8 with a(0)=1, a(1)=8 - Vincenzo Librandi, Jul 21 2010
a(n) = A191465(n)+1. - R. J. Mathar, Mar 10 2022

A155598 a(n) = 7^n-2^n+1.

Original entry on oeis.org

1, 6, 46, 336, 2386, 16776, 117586, 823416, 5764546, 40353096, 282474226, 1977324696, 13841283106, 96889002216, 678223056466, 4747561477176, 33232930504066, 232630513856136, 1628413597648306, 11398895184848856

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (10,-23,14).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597.

Table[7^n-2^n+1,{n,0,20}] (* or *) LinearRecurrence[{10,-23,14},{1,6,46},20] (* Harvey P. Dale, Feb 28 2013 *)

a(n)=7^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-7*x)-1/(1-2*x)+1/(1-x). E.g.f.: e^(7*x)-e^(2*x)+e^x.
a(n) = 9*a(n-1)-14*a(n-2)+6 with a(0)=1, a(1)=6 - Vincenzo Librandi, Jul 21 2010
a(0)=1, a(1)=6, a(2)=46, a(n) = 10*a(n-1)-23*a(n-2)+14*a(n-3). - Harvey P. Dale, Feb 28 2013
a(n) = A190540(n)+1. - R. J. Mathar, Mar 10 2022

A155601 a(n) = 10^n - 2^n + 1^n.

Original entry on oeis.org

1, 9, 97, 993, 9985, 99969, 999937, 9999873, 99999745, 999999489, 9999998977, 99999997953, 999999995905, 9999999991809, 99999999983617, 999999999967233, 9999999999934465, 99999999999868929, 999999999999737857, 9999999999999475713, 99999999999998951425

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (13,-32,20).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597, A155598, A155599, A155600.

Table[10^n-2^n+1,{n,0,20}] (* or *) LinearRecurrence[{13,-32,20},{1,9,97},20] (* Harvey P. Dale, Jan 13 2022 *)

a(n)=10^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-10*x)-1/(1-2*x)+1/(1-x).
E.g.f.: e^(10*x)-e^(2*x)+e^x.
a(n) = 12*a(n-1)-20*a(n-2)+9 with a(0)=1, a(1)=9. - Vincenzo Librandi, Jul 21 2010
a(n) = A060458(n)+1. - R. J. Mathar, Mar 10 2022

A155602 4^n + 3^n - 1.

Original entry on oeis.org

1, 6, 24, 90, 336, 1266, 4824, 18570, 72096, 281826, 1107624, 4371450, 17308656, 68703186, 273218424, 1088090730, 4338014016, 17309009346, 69106897224, 276040168410, 1102998412176, 4408506864306, 17623567104024, 70462887356490

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

Cf. A074501, A020515, A155588, A155590, A155592-A155594, A155596-A155601.

[(4^n + 3^n - 1): n in [0..30]]; // Vincenzo Librandi, Oct 17 2012

Table[4^n + 3^n - 1, {n, 0, 50}] (* Vincenzo Librandi, Oct 17 2012 *)
LinearRecurrence[{8,-19,12},{1,6,24},30] (* Harvey P. Dale, Apr 28 2018 *)

a(n)=4^n+3^n-1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-4*x)+1/(1-3*x)-1/(1-x). E.g.f.: e^(4*x)+e^(3*x)-e^x.
a(n) = 7*a(n-1) - 12*a(n-2) -6, n>1 - Gary Detlefs, Jun 21 2010
a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3), n>2, a(0)=1, a(1)=6, a(2)=24. - L. Edson Jeffery, Oct 17 2012
a(n) = A074605(n)-1. - R. J. Mathar, Mar 10 2022

A155603 a(n) = 5^n+3^n-1.

Original entry on oeis.org

1, 7, 33, 151, 705, 3367, 16353, 80311, 397185, 1972807, 9824673, 49005271, 244672065, 1222297447, 6108298593, 30531927031, 152630937345, 763068593287, 3815084686113, 19074648589591, 95370918425025, 476847618556327

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597, A155598, A155599, A155600, A155601, A155602.

Table[5^n+3^n-1,{n,0,30}] (* or *) LinearRecurrence[{9,-23,15},{1,7,33},30] (* Harvey P. Dale, Oct 14 2020 *)

a(n)=5^n+3^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-5*x)+1/(1-3*x)-1/(1-x).
E.g.f.: e^(5*x)+e^(3*x)-e^x.
a(n) = 8*a(n-1)-15*a(n-2)-8 with a(0)=1, a(1)=7. - Vincenzo Librandi, Jul 21 2010
a(n) = A074606(n)-1. - R. J. Mathar, Mar 10 2022

A155604 a(n) = 6^n + 3^n - 1.

Original entry on oeis.org

1, 8, 44, 242, 1376, 8018, 47384, 282122, 1686176, 10097378, 60525224, 362974202, 2177313776, 13062288338, 78368947064, 470199333482, 2821152954176, 16926788584898, 101560344088904, 609360902271962, 3656161926847376

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-27,18).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597, A155598, A155599, A155600, A155601, A155602, A155603.

Table[6^n + 3^n - 1, {n, 0, 25}] (* Paolo Xausa, Aug 21 2024 *)

a(n)=6^n+3^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-6*x)+1/(1-3*x)-1/(1-x).
E.g.f.: exp(6*x)+exp(3*x)-exp(x).
a(n) = 9*a(n-1)-18*a(n-2)-10 with a(0) = 1, a(1) = 8. - Vincenzo Librandi, Jul 21 2010

A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047

Offset: 1

Brad Clardy, Apr 01 2013

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. 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.
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's with the last difference in the pattern m. The number of ones in
   the pattern is 2^j - 1, where j is the column index.
As an example consider A(4) which is 9,
   the sequence B(n) where i XOR 8 = i - 8 starts as:
   8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
   with successive differences of:
   1, 1, 1, 1, 1, 1, 1, 9.
The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
The formula for diagonals above the main diagonal
  2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
  2^(2*n) - 2^(n  + (b/2)) + 1    n>=(b/2)+1 b=even number above diagonal
The formulas for diagonals below the main diagonal
  2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
  2^(2*n) - 2^(n - (b/2)) + 1       n>=(b/2)+1 b=even number below diagonal
Primes of this sequence are in A152449.

			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)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...

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

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

//program generates values in a table form
for i:=1 to 10 do
    m:=2^i - 1;
    m,[ m*2^n +1 : 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^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;

Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]

a(n) = A081118(n)+2; a(n)=(2^i-1)*2^j+1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

cp's OEIS Frontend

A155599 a(n) = 8^n - 2^n + 1^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A155596 a(n) = 5^n - 2^n + 1^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A155597 a(n) = 6^n - 2^n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A155600 a(n) = 9^n-2^n+1^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A155598 a(n) = 7^n-2^n+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A155601 a(n) = 10^n - 2^n + 1^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A155602 4^n + 3^n - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A155603 a(n) = 5^n+3^n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica