A163604 -id:A163604 - cp's OEIS frontend

A163605 a(n) = ((3+2sqrt(2))(5+sqrt(2))^n + (3-2sqrt(2))(5-sqrt(2))^n)/2.

3, 19, 121, 773, 4947, 31691, 203129, 1302397, 8352003, 53564899, 343552921, 2203536533, 14133648147, 90655141211, 581477504729, 3729706799437, 23923085385603, 153447597468979, 984245010820921, 6313155366422693

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

nonn

Binomial transform of A163604.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A163604.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+2*r)*(5+r)^n+(3-2*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 07 2009

LinearRecurrence[{10, -23}, {3, 19}, 50] (* G. C. Greubel, Jul 29 2017 *)

x='x+O('x^50); Vec((3-11*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 10*a(n-1)-23*a(n-2) for n > 1; a(0) = 3, a(1) = 19.
G.f.: (3-11*x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*( 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 07 2009

A163606 a(n) = ((3 + 2sqrt(2))(3 + sqrt(2))^n + (3 - 2sqrt(2))(3 - sqrt(2))^n)/2.

Original entry on oeis.org

3, 13, 57, 251, 1107, 4885, 21561, 95171, 420099, 1854397, 8185689, 36133355, 159500307, 704068357, 3107907993, 13718969459, 60558460803, 267317978605, 1179998646009, 5208766025819, 22992605632851, 101494271616373

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

nonn

Binomial transform of A048580. Inverse binomial transform of A163604.

For n >= 1, a(n-1) is the number of generalized compositions of n when there are 2^(i-1)+1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -7).

Cf. A048580, A163604.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+2*r)*(3+r)^n+(3-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 07 2009

LinearRecurrence[{6,-7},{3,13},40] (* Harvey P. Dale, Dec 24 2011 *)

x='x+O('x^50); Vec((3-5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 3, a(1) = 13.
G.f.: (3-5*x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*( 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 07 2009

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

cp's OEIS Frontend

A163605 a(n) = ((3+2sqrt(2))(5+sqrt(2))^n + (3-2sqrt(2))(5-sqrt(2))^n)/2.

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A163606 a(n) = ((3 + 2sqrt(2))(3 + sqrt(2))^n + (3 - 2sqrt(2))(3 - sqrt(2))^n)/2.

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

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cp's OEIS Frontend

A163605 a(n) = ((3+2*sqrt(2))*(5+sqrt(2))^n + (3-2*sqrt(2))*(5-sqrt(2))^n)/2.

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A163606 a(n) = ((3 + 2*sqrt(2))*(3 + sqrt(2))^n + (3 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.

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

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Magma

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A163605 a(n) = ((3+2sqrt(2))(5+sqrt(2))^n + (3-2sqrt(2))(5-sqrt(2))^n)/2.

A163606 a(n) = ((3 + 2sqrt(2))(3 + sqrt(2))^n + (3 - 2sqrt(2))(3 - sqrt(2))^n)/2.