A020727 -id:A020727 - cp's OEIS frontend

A021000 Duplicate of A020727.

2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048

Offset: 0

dead

A164090 a(n) = 2*a(n-2) for n > 2; a(1) = 2, a(2) = 3.

2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728

Offset: 1

Klaus Brockhaus, Aug 09 2009

Interleaving of A000079 without initial 1 and A007283.
Agrees from a(2) onward with A145751 for all terms listed there (up to 65536). Apparently equal to 2, 3 followed by A090989. Equals 2 followed by A163978.
Binomial transform is A000129 without first two terms, second binomial transform is A020727, third binomial transform is A164033, fourth binomial transform is A164034, fifth binomial transform is A164035.
Number of achiral necklaces or bracelets with n beads using up to 2 colors. For n=5, the eight achiral necklaces or bracelets are AAAAA, AAAAB, AAABB, AABAB, AABBB, ABABB, ABBBB, and BBBBB. - Robert A. Russell, Sep 22 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A029744, A145751, A090989, A163978, A000129, A020727, A164033, A164034, A164035, A052955, A027383, A060482, A070875.

Second column of A284855.

[ n le 2 select n+1 else 2*Self(n-2): n in [1..42] ];

a[n_] := If[EvenQ[n], 3*2^(n/2 - 1), 2^((n + 1)/2)]; Array[a, 42] (* Jean-François Alcover, Oct 12 2017 *)
RecurrenceTable[{a[1]==2,a[2]==3,a[n]==2a[n-2]},a,{n,50}] (* or *) LinearRecurrence[{0,2},{2,3},50] (* Harvey P. Dale, Mar 01 2018 *)

a(n) = if(n%2,2,3) * 2^((n-1)\2); \\ Andrew Howroyd, Oct 07 2017

a(n) = A029744(n+1).
a(n) = A052955(n-1) + 1.
a(n) = A027383(n-2) + 2 for n > 1.
a(n) = A060482(n-1) + 3 for n > 3.
a(n) = A070875(n) - A070875(n-1).
a(n) = (7 - (-1)^n)*2^((1/4)*(2*n - 1 + (-1)^n))/4.
G.f.: x*(2+3*x)/(1-2*x^2).
a(n) = A063759(n-1), n>1. - R. J. Mathar, Aug 17 2009
Sum_{n>=1} 1/a(n) = 5/3. - Amiram Eldar, Mar 28 2022

A056236 a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.

Original entry on oeis.org

2, 4, 12, 40, 136, 464, 1584, 5408, 18464, 63040, 215232, 734848, 2508928, 8566016, 29246208, 99852800, 340918784, 1163969536, 3974040576, 13568223232, 46324811776, 158162800640, 540001579008, 1843680714752, 6294719700992

Offset: 0

Henry Bottomley, Aug 11 2000

First differences give A060995. - Jeremy Gardiner, Aug 11 2013
Binomial transform of A002203 [Bhadouria].
The binomial transform of this sequence is 2, 6, 22, 90, 386, .. = 2*A083878(n). - R. J. Mathar, Nov 10 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Pooja Bhadouria, Deepika Jhala, and Bijendra Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, sequence B_2.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 9.
Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A006012, A007052, A020727.

LinearRecurrence[{4,-2},{2,4},30] (* Harvey P. Dale, Jan 18 2013 *)

PARI
```
a(n) = 2*real((2+quadgen(8))^n);
```

[lucas_number2(n,4,2) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

a(n) = 4*a(n-1) - 2*a(n-2).
a(n) = a(n-2) - a(n-1) + 2*A020727(n-1).
a(n) = 2*A006012(n) = 4*A007052(n-1).
For n>2, a(n) = floor((2+sqrt(2))*a(n-1)).
G.f.: 2*(1-2*x)/(1-4*x+2*x^2).
From L. Edson Jeffery, Apr 08 2011: (Start)
a(n) = 2^(2*n)*(cos(Pi/8)^(2*n) + cos(3*Pi/8)^(2*n)).
a(n) = 3*a(n-1) + Sum_{k=1..(n-2)} a(k), for n>1, with a(0)=2, a(1)=4. (End)
a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 8*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

More terms from James Sellers, Aug 25 2000

A164033 a(n) = ((4+3sqrt(2))(3+sqrt(2))^n + (4-3sqrt(2))(3-sqrt(2))^n)/4.

Original entry on oeis.org

2, 9, 40, 177, 782, 3453, 15244, 67293, 297050, 1311249, 5788144, 25550121, 112783718, 497851461, 2197622740, 9700776213, 42821298098, 189022355097, 834385043896, 3683153777697, 16258227358910, 71767287709581

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

nonn

Binomial transform of A020727. Third binomial transform of A164090. Inverse binomial transform of A164034.

Matthew House, Table of n, a(n) for n = 0..1542
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A020727, A164090 (2, 3, 4, 6, 8, 12), A164034.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(3+r)^n+(4-3*r)*(3-r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 09 2009

CoefficientList[Series[(2-3*x)/(1-6*x+7*x^2), {x, 0, 1000}],
  x] (* or *) LinearRecurrence[{6,-7},{2,9}, 50] (* G. C. Greubel, Sep 08 2017 *)

x='x+O('x^50); Vec((2-3*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 08 2017

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 2, a(1) = 9.
G.f.: (2-3*x)/(1-6*x+7*x^2).
E.g.f.: (2*cosh(sqrt(2)*x) + (3/sqrt(2))*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 09 2009

cp's OEIS Frontend

A021000 Duplicate of A020727.

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A164090 a(n) = 2*a(n-2) for n > 2; a(1) = 2, a(2) = 3.

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A056236 a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.

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A164033 a(n) = ((4+3*sqrt(2))*(3+sqrt(2))^n + (4-3*sqrt(2))*(3-sqrt(2))^n)/4.

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A164033 a(n) = ((4+3sqrt(2))(3+sqrt(2))^n + (4-3sqrt(2))(3-sqrt(2))^n)/4.