A013729 -id:A013729 - cp's OEIS frontend

A004171 a(n) = 2^(2n+1).

2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832, 140737488355328, 562949953421312

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(2, 8), L(2, 8), P(2, 8), T(2, 8). See A008776 for definitions of Pisot sequences.
In the Chebyshev polynomial of degree 2n, a(n) is the coefficient of x^2n. - Benoit Cloitre, Mar 13 2002
1/2 - 1/8 + 1/32 - 1/128 + ... = 2/5. - Gary W. Adamson, Mar 03 2009
From Adi Dani, May 15 2011: (Start)
Number of ways of placing an even number of indistinguishable objects in n + 1 distinguishable boxes with at most 3 objects in box.
Number of compositions of even natural numbers into n + 1 parts less than or equal to 3 (0 is counted as part). (End)
Also the number of maximal cliques in the (n+1)-Sierpinski tetrahedron graph for n > 0. - Eric W. Weisstein, Dec 01 2017
Assuming the Collatz conjecture is true, any starting number eventually leads to a power of 2. A number in this sequence can never be the first power of 2 in a Collatz sequence except of course for the Collatz sequence starting with that number. For example, except for 8, 4, 2, 1, any Collatz sequence that includes 8 must also include 16 (e.g., 5, 16, 8, 4, 2, 1). - Alonso del Arte, Oct 01 2019
First differences of A020988, and thus the "wavelengths" of the local maxima in A020986. See the Brillhart and Morton link, pp. 855-856. - John Keith, Mar 04 2021

			G.f. = 2 + 8*x + 32*x^2 + 128*x^3 + 512*x^4 + 2048*x^5 + 8192*x^6 + 32768*x^7 + ...
From _Adi Dani_, May 15 2011: (Start)
a(1) = 8 because all compositions of even natural numbers into 2 parts less than or equal to 3 are:
  for 0: (0, 0)
  for 2: (0, 2), (2, 0), (1, 1)
  for 4: (1, 3), (3, 1), (2, 2)
  for 6: (3, 3).
a(2) = 32 because all compositions of even natural numbers into 3 parts less than or equal to 3 are:
  for 0: (0, 0, 0)
  for 2: (0, 0, 2), (0, 2, 0), (2, 0, 0), (0, 1, 1), (1, 0, 1) , (1, 1, 0)
  for 4: (0, 1, 3), (0, 3, 1), (1, 0, 3), (1, 3, 0), (3, 0, 1), (3, 1, 0), (0, 2, 2), (2, 0, 2), (2, 2, 0), (1, 1, 2), (1, 2, 1), (2, 1, 1)
  for 6: (0, 3, 3), (3, 0, 3), (3, 3, 0), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), (2, 2, 2)
  for 8: (2, 3, 3), (3, 2, 3), (3, 3, 2).
(End)

Adi Dani, Quasicompositions of natural numbers, Proceedings of III congress of mathematicians of Macedonia, Struga Macedonia 29 IX -2 X 2005 pages 225-238.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023).
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Tanya Khovanova, Recursive Sequences.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Index entries for linear recurrences with constant coefficients, signature (4).
Index to divisibility sequences.

Cf. A013708-A013729.
Absolute value of A009117. Essentially the same as A081294.
Cf. A132020, A164632. Equals A000980(n) + 2*A181765(n). Cf. A013776.

List([0..30],n->2^(2*n+1)); # Muniru A Asiru, Mar 12 2019

a004171 = (* 2) . a000302
a004171_list = iterate (* 4) 2  -- Reinhard Zumkeller, Jan 09 2013

[2^(2*n+1): n in [0..30]]; // Vincenzo Librandi, May 16 2011

seq(2^(2*n+1),n=0..24); # Nathaniel Johnston, Jun 25 2011

Table[2^(2 n + 1), {n, 0, 24}]
2^(2 Range[20] - 1) (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4}, {2}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[2/(1 - 4 x), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=2<<(2*n) \\ Charles R Greathouse IV, Apr 07 2012

a(n) = 2^(2*n+1) \\ Michel Marcus, Aug 12 2014

[2**(2*n+1) for n in range(0,25)] # Stefano Spezia, Jul 23 2025

((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).map(2 * ) // _Alonso del Arte, Sep 12 2019

a(n) = 2*4^n.
a(n) = 4*a(n-1).
1 = 1/2 + Sum_{n >= 1} 3/a(n) = 3/6 + 3/8 + 3/32 + 3/128 + 3/512 + 3/2048 + ...; with partial sums: 1/2, 31/32, 127/128, 511/512, 2047/2048, ... - Gary W. Adamson, Jun 16 2003
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 2*A000302(n).
G.f.: 2/(1-4*x). (End)
a(n) = A081294(n+1) = A028403(n+1) - A000079(n+1) for n >= 1. a(n-1) = A028403(n) - A000079(n). - Jaroslav Krizek, Jul 27 2009
E.g.f.: 2*exp(4*x). - Ilya Gutkovskiy, Nov 01 2016
a(n) = A002063(n)/3 - A000302(n). - Zhandos Mambetaliyev, Nov 19 2016
a(n) = Sum_{k = 0..2*n} (-1)^(k+n)*binomial(4*n + 2, 2*k + 1); a(2*n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A013776(n). - Peter Bala, Nov 25 2016
Product_{n>=0} (1 - 1/a(n)) = A132020. - Amiram Eldar, May 08 2023

A013715 a(n) = 10^(2*n+1).

Original entry on oeis.org

10, 1000, 100000, 10000000, 1000000000, 100000000000, 10000000000000, 1000000000000000, 100000000000000000, 10000000000000000000, 1000000000000000000000, 100000000000000000000000, 10000000000000000000000000, 1000000000000000000000000000, 100000000000000000000000000000

Offset: 0

N. J. A. Sloane

Bisection of A011557 (powers of 10). - Michel Marcus, Jan 17 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (100).

Cf. A013708-A013714, A013716-A013729, A011557.

Cf. A005408, A098608, A013747.

[10^(2*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 26 2011

seq(10^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

10^(2 Range[0, 15] + 1) (* Wesley Ivan Hurt, Jan 17 2016 *)

a(n) = 10^(2*n+1); \\ Michel Marcus, Jan 17 2016

Vec(10/(1-100*x) + O(x^100)) \\ Altug Alkan, Jan 17 2016

From Philippe Deléham, Nov 25 2008: (Start)
G.f.: 10/(1-100*x).
a(n) = 100*a(n-1), n>0; a(0)=10. (End)
From Elmo R. Oliveira, Aug 26 2024 (Start)
E.g.f.: 10*exp(100*x).
a(n) = 10*A098608(n) = A011557(A005408(n)) = A013747(n)/10^(n+1). (End)

A013716 a(n) = 11^(2*n + 1).

Original entry on oeis.org

11, 1331, 161051, 19487171, 2357947691, 285311670611, 34522712143931, 4177248169415651, 505447028499293771, 61159090448414546291, 7400249944258160101211, 895430243255237372246531

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 (121).

Bisection of A001020 (11^n).

Cf. A013708-A013715, A013717-A013729.

[11^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

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

11^(2*Range[0,20]+1) (* or *) NestList[121#&,11,20] (* Harvey P. Dale, Jun 21 2024 *)

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

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 121*a(n-1), a(0)=11.
G.f.: 11/(1-121*x). (End)

A013718 a(n) = 13^(2*n + 1).

Original entry on oeis.org

13, 2197, 371293, 62748517, 10604499373, 1792160394037, 302875106592253, 51185893014090757, 8650415919381337933, 1461920290375446110677, 247064529073450392704413, 41753905413413116367045797

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 (169).

Bisection of A001022 (12^n).

Cf. A013708-A013717, A013719-A013729.

[13^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(13^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

a(n)=13^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 169*a(n-1); a(0)=13.
G.f.: 13/(1-169*x). (End)

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

Original entry on oeis.org

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)

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)

cp's OEIS Frontend

A004171 a(n) = 2^(2n+1).

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A013715 a(n) = 10^(2*n+1).

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A013716 a(n) = 11^(2*n + 1).

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A013718 a(n) = 13^(2*n + 1).

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

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

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