A083882 -id:A083882 - cp's OEIS frontend

A083881 a(n) = 6a(n-1) - 6a(n-2), with a(0)=1, a(1)=3.

1, 3, 12, 54, 252, 1188, 5616, 26568, 125712, 594864, 2814912, 13320288, 63032256, 298271808, 1411437312, 6678993024, 31605334272, 149558047488, 707716279296, 3348949390848, 15847398669312, 74990695670784, 354859782008832

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A001075.

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

Cf. A083882.

Cf. A030192.

a:=[1,3];; for n in [3..30] do a[n]:=6*a[n-1]-6*a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,3]; [n le 2 select I[n] else 6*Self(n-1) -6*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019

f[n_]:= Simplify[(3 + Sqrt@3)^n + (3 - Sqrt@3)^n]/2; Array[f, 30, 0] (* Robert G. Wilson v, Oct 31 2010 *)
LinearRecurrence[{6,-6}, {1,3}, 30] (* G. C. Greubel, Aug 01 2019 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-6*x+6*x^2)) \\ G. C. Greubel, Aug 01 2019

((1-3*x)/(1-6*x+6*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = ((3-sqrt(3))^n + (3+sqrt(3))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*3^(n-k).
G.f.: (1-3*x)/(1-6*x+6*x^2).
E.g.f.: exp(3*x) * cosh(x*sqrt(3)).
a(n) = right and left terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 4 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A030192(n) a(n)]. E.g. a(3) = 54 since M^3 * [1 1 1] = [54 144 54] = [a(3) A030192(3) a(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = Sum_{k, 0<=k<=n}3^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
G.f.: A(x) = G(0) where G(k) =  1 + 3*x/((1-3*x) - x*(1-3*x)/(x + (1-3*x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 29 2012.

A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.
First differences are in A161728.
Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ];  // Klaus Brockhaus, Dec 31 2008

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Join[{a=1,b=8},Table[c=8*b-13*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-13},{1,8},40] (* Harvey P. Dale, Aug 16 2012 *)

a(n)=([0,1; -13,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 04 2016

[lucas_number1(n,8,13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: sinh(sqrt(3)*x)*exp(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016
a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016
a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

A143647 a(n) = ((5 + sqrt(3))^n + (5 - sqrt(3))^n)/2.

Original entry on oeis.org

1, 5, 28, 170, 1084, 7100, 47152, 315320, 2115856, 14221520, 95666368, 643790240, 4333242304, 29169037760, 196359046912, 1321871638400, 8898817351936, 59906997474560, 403295993003008, 2715005985589760, 18277548009831424

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Oct 27 2008

nonn

Binomial transform of A083882. - R. J. Mathar, Nov 01 2008

Inverse binomial transform of A147961.

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

Cf. A083882, A147961, A098158.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r3)^n+(5-r3)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 01 2008

Simplify[With[{c=Sqrt[3]},Table[((5+c)^n+(5-c)^n)/2,{n,0,25}]]] (* or *) LinearRecurrence[{10,-22},{1,5},25] (* Harvey P. Dale, Jun 04 2011 *)

From Philippe Deléham, Klaus Brockhaus and R. J. Mathar, Nov 01 2008: (Start)
a(n) = 10*a(n-1) - 22*a(n-2), a(0)=1, a(1)=5.
G.f.: (1-5x)/(1-10x+22*x^2). (End)
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2*k)*3^(n-k))/5^n. - Philippe Deléham, Nov 06 2008

More terms from Klaus Brockhaus and R. J. Mathar, Nov 01 2008

Edited by Klaus Brockhaus, Jul 15 2009

A164310 a(n) = 6a(n-1) - 6a(n-2) for n > 1; a(0) = 4, a(1) = 15.

Original entry on oeis.org

4, 15, 66, 306, 1440, 6804, 32184, 152280, 720576, 3409776, 16135200, 76352544, 361304064, 1709709120, 8090430336, 38284327296, 181163381760, 857274326784, 4056665670144, 19196348060160, 90838094340096, 429850477679616

Offset: 0

Klaus Brockhaus, Aug 12 2009

Binomial transform of A077236. Inverse binomial transform of A083882 without initial 1.

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

Cf. A077236, A083882.

[ n le 2 select 11*n-7 else 6*Self(n-1)-6*Self(n-2): n in [1..22] ];

LinearRecurrence[{6,-6}, {4,15}, 50] (* or *) CoefficientList[Series[(4 - 9*x)/(1 - 6*x + 6*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 13 2017 *)

x='x+O('x^50); Vec((4-9*x)/(1-6*x+6*x^2)) \\ G. C. Greubel, Sep 13 2017

a(n) = ((4+sqrt(3))*(3+sqrt(3))^n + (4-sqrt(3))*(3-sqrt(3))^n)/2.
G.f.: (4-9*x)/(1-6*x+6*x^2).
E.g.f.: (4*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x))*exp(3*x). - G. C. Greubel, Sep 13 2017

cp's OEIS Frontend

A083881 a(n) = 6*a(n-1) - 6*a(n-2), with a(0)=1, a(1)=3.

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A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

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Magma

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A143647 a(n) = ((5 + sqrt(3))^n + (5 - sqrt(3))^n)/2.

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Author

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Programs

Magma

Mathematica

Formula

Extensions

A164310 a(n) = 6*a(n-1) - 6*a(n-2) for n > 1; a(0) = 4, a(1) = 15.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A083881 a(n) = 6a(n-1) - 6a(n-2), with a(0)=1, a(1)=3.

A164310 a(n) = 6a(n-1) - 6a(n-2) for n > 1; a(0) = 4, a(1) = 15.