A055845 -id:A055845

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A153597

a(n) = ((6 + sqrt(3))^n - (6 - sqrt(3))^n)/(2*sqrt(3)).

+10
2

1, 12, 111, 936, 7569, 59940, 469503, 3656016, 28378593, 219894588, 1702241487, 13170376440, 101870548209, 787824155988, 6092161780959, 47107744223904, 364251591915201, 2816463543593580, 21777259989921327, 168383822940467784

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Fourth binomial transform of A055845.

lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(3) = 7.73205080756887729....

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (12,-33).

FORMULA

G.f.: x/(1 - 12*x + 33*x^2). - Klaus Brockhaus, Dec 31 2008, (corrected Oct 11 2009)

a(n) = 12*a(n-1) - 33*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(6*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

MATHEMATICA

LinearRecurrence[{12, -33}, {1, 12}, 25] (* G. C. Greubel, Aug 22 2016 *)

PROG

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

(Sage) [lucas_number1(n, 12, 33) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

(Magma) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-33*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

CROSSREFS

Cf. A002194 (decimal expansion of sqrt(3)), A055845.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

Edited by Klaus Brockhaus, Oct 11 2009

STATUS

approved

A166450

a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

+10
1

1, 6, 3, 18, 9, 54, 27, 162, 81, 486, 243, 1458, 729, 4374, 2187, 13122, 6561, 39366, 19683, 118098, 59049, 354294, 177147, 1062882, 531441, 3188646, 1594323, 9565938, 4782969, 28697814, 14348907, 86093442, 43046721, 258280326

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Interleaving of A000244 and 6*A000244.

Second binomial transform is A055845. Sixth binomial transform is A153597.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,3).

FORMULA

a(n) = (3+(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2.

G.f.: x*(1+6*x)/(1-3*x^2).

MATHEMATICA

LinearRecurrence[{0, 3}, {1, 6}, 50] (* G. C. Greubel, May 14 2016 *)

PROG

(Magma) [ n le 2 select 5*n-4 else 3*Self(n-2): n in [1..35] ];

CROSSREFS

Cf. A000244 (powers of 3), A055845, A153597.

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Oct 13 2009

STATUS

approved

A257487

Expansion of ( -4+15*x-8*x^2 ) / ( (x-1)*(x^2-4*x+1) ).

+10
1

4, 5, 13, 44, 160, 593, 2209, 8240, 30748, 114749, 428245, 1598228, 5964664, 22260425, 83077033, 310047704, 1157113780, 4318407413, 16116515869, 60147656060, 224474108368, 837748777409, 3126521001265, 11668335227648, 43546819909324

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..24.

M. Knor, R. Skrekovski, Wiener index of generalized 4-stars and their quadratic line graphs, Austral. J. Combinat. 58 (1) (2014) 119-126, eq. (11).

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

FORMULA

a(n) = (5*A001353(n+1)-13*A001353(n)+3)/2. - R. J. Mathar, May 26 2016

MAPLE

A257487 := proc(n)

(5+sqrt(3))/4*(2-sqrt(3))^n+(5-sqrt(3))/4*(2+sqrt(3))^n+3/2 ;

expand(%) ;

end proc:

seq(A257487(n), n=0..30) ;

PROG

(PARI) Vec(( -4+15*x-8*x^2 ) / ( (x-1)*(x^2-4*x+1) ) + O(x^50)) \\ Michel Marcus, Apr 26 2015

CROSSREFS

Cf. A055845 (first differences).

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Apr 26 2015

STATUS

approved

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