A013708 -id:A013708 - cp's OEIS frontend

A013722 a(n) = 17^(2*n + 1).

17, 4913, 1419857, 410338673, 118587876497, 34271896307633, 9904578032905937, 2862423051509815793, 827240261886336764177, 239072435685151324847153, 69091933913008732880827217

Offset: 0

N. J. A. Sloane

Sum_{n>=0} 1/a(n) = 17/288. - Jaume Oliver Lafont, Feb 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (289).

Bisection of A001026 (17^n).

Cf. A013708-A013721, A013723-A013729.

[17^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(17^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

17^(2*Range[0,10]+1) (* or *) NestList[289#&,17,10] (* Harvey P. Dale, Aug 15 2012 *)

a(n)=17^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 289*a(n-1), a(0)=17.
G.f.: 17/(1-289*x). (End)

A013724 a(n) = 19^(2*n + 1).

Original entry on oeis.org

19, 6859, 2476099, 893871739, 322687697779, 116490258898219, 42052983462257059, 15181127029874798299, 5480386857784802185939, 1978419655660313589123979, 714209495693373205673756419

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (361).

Bisection of A001029.

Cf. A013708-A013723, A013725-A013729.

[19^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(19^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

NestList[361#&,19,20] (* Harvey P. Dale, Jun 12 2017 *)

a(n)=19^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 361*a(n-1); a(0)=19.
G.f.: 19/(1-361*x). (End)

A013728 a(n) = 23^(2*n + 1).

Original entry on oeis.org

23, 12167, 6436343, 3404825447, 1801152661463, 952809757913927, 504036361936467383, 266635235464391245607, 141050039560662968926103, 74615470927590710561908487, 39471584120695485887249589623

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (529).

Cf. A013708-A013727, A013729.

[23^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(23^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 26 2011

NestList[529#&,23,15] (* Harvey P. Dale, Jan 17 2012 *)

a(n)=23^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 529*a(n-1); a(0)=23.
G.f.: 23/(1-529*x). (End)

A064932 Period of the continued fraction for sqrt(3^(2n+1)).

Original entry on oeis.org

2, 10, 30, 98, 270, 818, 2382, 7282, 21818, 65650, 196406, 589982, 1768938, 5309294, 15924930, 47779238, 143322850, 429998586, 1289970842, 3869957114, 11609762666, 34829416842, 104488103446

Offset: 1

Wouter Meeussen, Oct 26 2001

Limit_{n->oo} a(n)/3^n = 1.11... - A.H.M. Smeets, Oct 25 2017

R. K. Guy, personal communication.

Cf. A003285, A013708, A059927.

Table[Length[Last[ContinuedFraction[Sqrt[3^(2n+1)] ]]], {n, 10}]

a(n) = A003285(A013708(n)). - Michel Marcus, Sep 25 2019

a(11)-a(13) from A.H.M. Smeets, Sep 28 2017
a(14) from A.H.M. Smeets, Oct 08 2017
a(15)-a(19) from Daniel Suteu, Jan 24 2019
a(20) from Chai Wah Wu, Sep 23 2019
a(21)-a(23) from Chai Wah Wu, Sep 25 2019

A262715 a(n) = 29^(2*n+1).

Original entry on oeis.org

29, 24389, 20511149, 17249876309, 14507145975869, 12200509765705829, 10260628712958602189, 8629188747598184440949, 7257147736730073114838109, 6103261246589991489578849669, 5132842708382182842735812571629, 4316720717749415770740818372739989

Offset: 0

Vincenzo Librandi, Oct 02 2015

29*a(n) is a square.

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (841).

Second bisection of A009973 (powers of 29).

Cf. similar sequences of the form p^(2*n+1), with p prime: A004171 (p=2), A013708 (p=3), A013710 (p=5), A013712 (p=7), A013716 (p=11), A013718 (p=13), A013722 (p=17), A013724 (p=19), A013728 (p=23), this sequence (p=29), A262716 (p=31), A262786 (p=37), A262787 (p=41), A155477 (p=43).

Magma
```
[29^(2*n+1): n in [0..15]];
```

29^Range[1, 30, 2]
NestList[841#&,29,20] (* Harvey P. Dale, May 16 2025 *)

vector(20, n, n--; 29^(2*n+1)) \\ Altug Alkan, Oct 02 2015

G.f.: 29/(1 - 841*x).
a(n) = 841*a(n-1).
Sum_{i>=0} (-1)^i/a(i) = 29*A021846; Sum_{i>=0} 1/a(i) = 2.9*A021088. [Bruno Berselli, Oct 06 2015]

A013711 a(n) = 6^(2*n+1).

Original entry on oeis.org

6, 216, 7776, 279936, 10077696, 362797056, 13060694016, 470184984576, 16926659444736, 609359740010496, 21936950640377856, 789730223053602816, 28430288029929701376, 1023490369077469249536, 36845653286788892983296, 1326443518324400147398656, 47751966659678405306351616

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (36).

Cf. A000400, A004171, A005408, A009980, A013708.

[6^(2*n+1): n in [0..20]]; // Vincenzo Librandi, May 26 2011

lst={}; Do[AppendTo[lst,6^(2*n+1)],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
NestList[36#&,6,20] (* Harvey P. Dale, Oct 18 2012 *)

a(n)=6^(2*n+1) \\ Charles R Greathouse IV, Jul 10 2016

G.f.: 6/(1-36*x). - Philippe Deléham, Nov 24 2008
a(n) = 36*a(n-1), a(0)=6. - Vincenzo Librandi, May 26 2011
From Elmo R. Oliveira, Aug 26 2024: (Start)
E.g.f.: 6*exp(36*x).
a(n) = 6*A009980(n) = A004171(n)*A013708(n) = A000400(A005408(n)). (End)

A013717 a(n) = 12^(2*n + 1).

Original entry on oeis.org

12, 1728, 248832, 35831808, 5159780352, 743008370688, 106993205379072, 15407021574586368, 2218611106740436992, 319479999370622926848, 46005119909369701466112, 6624737266949237011120128

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (144).

Bisection of A001021.

Cf. A013708-A013716, A013718-A013729.

[12^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(12^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

a(n)=12^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 144*a(n-1), a(0)=12.
G.f.: 12/(1-144*x). (End)

A013723 a(n) = 18^(2*n + 1).

Original entry on oeis.org

18, 5832, 1889568, 612220032, 198359290368, 64268410079232, 20822964865671168, 6746640616477458432, 2185911559738696531968, 708235345355337676357632, 229468251895129407139872768

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (324).

Bisection of A001027 (18^n).

Cf. A013708-A013722, A013724-A013729.

[18^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(18^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

18^(2*Range[0,20]+1) (* or *) NestList[324#&,18,20] (* Harvey P. Dale, Sep 28 2021 *)

a(n)=18^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 324*a(n-1); a(0)=18.
G.f.: 18/(1-324*x). (End)

A013725 a(n) = 20^(2*n + 1).

Original entry on oeis.org

20, 8000, 3200000, 1280000000, 512000000000, 204800000000000, 81920000000000000, 32768000000000000000, 13107200000000000000000, 5242880000000000000000000, 2097152000000000000000000000, 838860800000000000000000000000, 335544320000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (400).

Bisection of A009964 (20^n).

Cf. A004171, A005408, A013708-A013724, A013726-A013729.

[20^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(20^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

NestList[400*# &, 20, 15] (* Paolo Xausa, Jul 12 2025 *)

a(n)=20^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 400*a(n-1); a(0)=20.
G.f.: 20/(1-400*x). (End)
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: 20*exp(400*x).
a(n) = A004171(n)*A013715(n) = A009964(A005408(n)). (End)

A013726 a(n) = 21^(2*n + 1).

Original entry on oeis.org

21, 9261, 4084101, 1801088541, 794280046581, 350277500542221, 154472377739119461, 68122318582951682301, 30041942495081691894741, 13248496640331026125580781, 5842587018385982521381124421, 2576580875108218291929075869661, 1136272165922724266740722458520501

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (441).

Bisection of A009965 (21^n).

Cf. A005408, A013708-A013725, A013727-A013729.

[21^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(21^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

Table[21^(2 n + 1), {n, 0, 15}] (* Wesley Ivan Hurt, Feb 10 2014 *)

a(n)=21^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 441*a(n-1); a(0)=21.
G.f.: 21/(1-441*x). (End)
a(n) = A009965(A005408(n)). - Wesley Ivan Hurt, Feb 10 2014
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: 21*exp(441*x).
a(n) = A013708(n)*A013712(n). (End)

cp's OEIS Frontend

A013722 a(n) = 17^(2*n + 1).

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A013724 a(n) = 19^(2*n + 1).

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A013728 a(n) = 23^(2*n + 1).

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A064932 Period of the continued fraction for sqrt(3^(2n+1)).

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A262715 a(n) = 29^(2*n+1).

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A013711 a(n) = 6^(2*n+1).

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A013717 a(n) = 12^(2*n + 1).

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