A162557 -id:A162557 - cp's OEIS frontend

A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.

1, 6, 38, 248, 1644, 10984, 73672, 495072, 3329936, 22407776, 150819168, 1015220608, 6834184384, 46006990464, 309717848192, 2085024691712, 14036454256896, 94493999351296, 636137999861248, 4282512012883968

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Fifth binomial transform of A108411. Binomial transform of A162557. Inverse binomial transform of A162757.

2nd binomial transform of A086405. - R. J. Mathar, Jul 17 2009

Index entries for linear recurrences with constant coefficients, signature (10, -22).

Cf. A108411 (powers of 3 repeated), A162557, A162757.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1-4*x)/(1-10*x+22*x^2).
From R. J. Mathar, Jul 17 2009: (Start)
a(n) = 10*a(n-2) - 22*a(n-2).
G.f.: (1-4*x)/(1-10*x+22*x^2). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 13 2009

More terms from R. J. Mathar, Jul 17 2009

A163470 a(n) = 8a(n-1) - 13a(n-2) for n > 1; a(0) = 3, a(1) = 15.

Original entry on oeis.org

3, 15, 81, 453, 2571, 14679, 84009, 481245, 2757843, 15806559, 90600513, 519318837, 2976744027, 17062807335, 97804786329, 560621795277, 3213512139939, 18420013780911, 105584452428081, 605215440272805

Offset: 0

Klaus Brockhaus, Aug 11 2009

Binomial transform of A083881 without initial 1. Inverse binomial transform of A163471.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A083881, A163471.

[ n le 2 select 12*n-9 else 8*Self(n-1)-13*Self(n-2): n in [1..22] ];

LinearRecurrence[{8, -13}, {3, 15}, 50] (* G. C. Greubel, Jul 25 2017 *)

x='x+O('x^50); Vec((3-9*x)/(1-8*x+13*x^2)) \\ G. C. Greubel, Jul 25 2017

a(n) = ((3+sqrt(3))*(4+sqrt(3))^n + (3-sqrt(3))*(4-sqrt(3))^n)/2.
G.f.: (3-9*x)/(1-8*x+13*x^2).
a(n) = 3*A162557(n). - R. J. Mathar, Jun 14 2016
E.g.f.: (1/2)*exp(4*x)*(6*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x)). - G. C. Greubel, Jul 25 2017

cp's OEIS Frontend

A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.

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A163470 a(n) = 8a(n-1) - 13a(n-2) for n > 1; a(0) = 3, a(1) = 15.

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cp's OEIS Frontend

A162558 a(n) = ((3+sqrt(3))*(5+sqrt(3))^n + (3-sqrt(3))*(5-sqrt(3))^n)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

A163470 a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0) = 3, a(1) = 15.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.

A163470 a(n) = 8a(n-1) - 13a(n-2) for n > 1; a(0) = 3, a(1) = 15.