A241204 -id:A241204 - cp's OEIS frontend

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

1, 10, 38, 132, 452, 1544, 5272, 18000, 61456, 209824, 716384, 2445888, 8350784, 28511360, 97343872, 332352768, 1134723328, 3874187776, 13227304448, 45160842240, 154188760064, 526433355776, 1797355902976, 6136556900352, 20951515795456, 71532949381120

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Binomial transform of A048696. Second binomial transform of A164587. Inverse binomial transform of A164299.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Sequences in the class a(n, m): this sequence (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
Cf. A007070, A016921, A048696, A056236, A112032, A164587, A241204.
Cf. A016116(n+1).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+r)^n+(1-4*r)*(2-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+6*x)/(1-4*x+2*x^2) )); // G. C. Greubel, Dec 14 2018

a:=n->((1+4*sqrt(2))*(2+sqrt(2))^n+(1-4*sqrt(2))*(2-sqrt(2))^n)/2: seq(floor(a(n)),n=0..25); # Muniru A Asiru, Dec 15 2018

LinearRecurrence[{4,-2}, {1,10}, 50] (* or *) CoefficientList[Series[(1 + 6*x)/(1 - 4*x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+6*x)/(1-4*x+2*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 14 2018; Mar 12 2021

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0)=1, a(1)=10.
G.f.: (1+6*x)/(1-4*x+2*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = A056236(n) + 8*A007070(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A305031 Expansion of ((1 + 2x)/(1 - 2x))^(3/2).

Original entry on oeis.org

1, 6, 18, 44, 102, 228, 500, 1080, 2310, 4900, 10332, 21672, 45276, 94248, 195624, 404976, 836550, 1724580, 3549260, 7293000, 14965236, 30669496, 62783448, 128388624, 262303132, 535422888, 1092063000, 2225728400, 4533175800, 9226818000, 18769219920, 38158909920

Offset: 0

Seiichi Manyama, May 24 2018

nonn

Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ... then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..3000

((1 + 2*x)/(1 - 2*x))^(m/2): A063886 (m=1), this sequence (m=3), A241204 (m=4).

Cf. A304941, A305032.

[n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 2*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023

CoefficientList[Series[((1+2*x)/(1-2*x))^(3/2), {x,0,40}], x] (* G. C. Greubel, Jun 07 2023 *)

N=66; x='x+O('x^N); Vec(((1+2*x)/(1-2*x))^(3/2))

@CachedFunction
def a(n): # b = A305031
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 2*(n-2)*a(n-2))//n
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

n*a(n) = 6*a(n-1) + 4*(n-2)*a(n-2) for n > 1.

a(n) ~ 2^(n + 5/2) * sqrt(n/Pi). - Vaclav Kotesovec, May 28 2018

A236967 Expansion of (1+3x)^2/(1-3x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

cp's OEIS Frontend

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

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A305031 Expansion of ((1 + 2x)/(1 - 2x))^(3/2).

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A236967 Expansion of (1+3x)^2/(1-3x)^2.

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cp's OEIS Frontend

A164298 a(n) = ((1+4*sqrt(2))*(2+sqrt(2))^n + (1-4*sqrt(2))*(2-sqrt(2))^n)/2.

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Magma

Magma

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A305031 Expansion of ((1 + 2*x)/(1 - 2*x))^(3/2).

Views

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Magma

Mathematica

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SageMath

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A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

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Extensions

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

A305031 Expansion of ((1 + 2x)/(1 - 2x))^(3/2).

A236967 Expansion of (1+3x)^2/(1-3x)^2.