A145751 -id:A145751 - cp's OEIS frontend

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

A063759 Spherical growth series for modular group.

Original entry on oeis.org

1, 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

Offset: 0

N. J. A. Sloane, Aug 14 2001

Also number of sequences S of length n with entries in {1,..,q} where q = 3, satisfying the condition that adjacent terms differ in absolute value by exactly 1, see examples. - W. Edwin Clark, Oct 17 2008

			For n = 2 the a(2) = 4 sequences are (1,2),(2,1),(2,3),(3,2). - _W. Edwin Clark_, Oct 17 2008
From _Joerg Arndt_, Nov 23 2012: (Start)
There are a(6) = 16 such words of length 6:
[ 1]   [ 1 2 1 2 1 2 ]
[ 2]   [ 1 2 1 2 3 2 ]
[ 3]   [ 1 2 3 2 1 2 ]
[ 4]   [ 1 2 3 2 3 2 ]
[ 5]   [ 2 1 2 1 2 1 ]
[ 6]   [ 2 1 2 1 2 3 ]
[ 7]   [ 2 1 2 3 2 1 ]
[ 8]   [ 2 1 2 3 2 3 ]
[ 9]   [ 2 3 2 1 2 1 ]
[10]   [ 2 3 2 1 2 3 ]
[11]   [ 2 3 2 3 2 1 ]
[12]   [ 2 3 2 3 2 3 ]
[13]   [ 3 2 1 2 1 2 ]
[14]   [ 3 2 1 2 3 2 ]
[15]   [ 3 2 3 2 1 2 ]
[16]   [ 3 2 3 2 3 2 ]
(End)

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to modular groups
Index entries for linear recurrences with constant coefficients, signature (0,2)

Cf. A054886, A029744.

The sequence (ternary strings) seems to be related to A029744 and A090989.

import Data.List (transpose)
a063759 n = a063759_list !! n
a063759_list = concat $ transpose [a151821_list, a007283_list]
-- Reinhard Zumkeller, Dec 16 2013

CoefficientList[Series[(1+3*x+2*x^2)/(1-2*x^2),{x,0,40}],x](* Jean-François Alcover, Mar 21 2011 *)
Join[{1},Transpose[NestList[{Last[#],2First[#]}&,{3,4},40]][[1]]] (* Harvey P. Dale, Oct 22 2011 *)

a(n)=([0,1; 2,0]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Feb 09 2017

G.f.: (1+3*x+2*x^2)/(1-2*x^2).
a(n) = 2*a(n-2), n>2. - Harvey P. Dale, Oct 22 2011
a(2*n) = A151821(n+1); a(2*n+1) = A007283(n). - Reinhard Zumkeller, Dec 16 2013

Information from A145751 included by Joerg Arndt, Dec 03 2012

A163978 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 4.

Original entry on oeis.org

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 07 2009

Interleaving of A007283 and A000079 without initial terms 1 and 2.
Equals A029744 without first two terms. Agrees with A145751 for all terms listed there (up to 65536).
Binomial transform is A078057 without initial 1, second binomial transform is A048580, third binomial transform is A163606, fourth binomial transform is A163604, fifth binomial transform is A163605.
a(n) is the number of vertices of the (n-1)-iterated line digraph L^{n-1}(G) of the digraph G(=L^0(G)) with vertices u,v,w and arcs u->v, v->u, v->w, w->v. - Miquel A. Fiol, Jun 08 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Miquel A. Fiol, J. L. A. Yebra, and I. Alegre, Line digraph iterations and the (d,k) digraph problem, IEEE Trans. Comput. C-33(5) (1984), 400-403.
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A027383, A029744, A048580, A052955, A063759, A078057, A090989, A145751, A163604, A163605, A163606.

[ n le 2 select n+2 else 2*Self(n-2): n in [1..41] ];

LinearRecurrence[{0,2}, {3,4}, 52] (* or *) Table[(1/2)*(5-(-1)^n )*2^((2*n-1+(-1)^n)/4), {n,50}] (* G. C. Greubel, Aug 24 2017 *)

my(x='x+O('x^50)); Vec(x*(3+4*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 24 2017

[(2+(n%2))*2^((n-(n%2))//2) for n in range(1,41)] # G. C. Greubel, Jun 13 2024

a(n) = A027383(n-1) + 2.
a(n) = A052955(n) + 1 for n >= 1.
a(n) = (1/2)*(5 - (-1)^n)*2^((2*n - 1 + (-1)^n)/4).
G.f.: x*(3+4*x)/(1-2*x^2).
a(n) = A090989(n-1).
E.g.f.: (1/2)*(4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 4). - G. C. Greubel, Aug 24 2017
a(n) = A063759(n), n >= 1. - R. J. Mathar, Jan 25 2023

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

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A063759 Spherical growth series for modular group.

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A163978 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 4.

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