The On-Line Encyclopedia of Integer Sequences (OEIS)

A153594 revision #41

A153594

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

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

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OFFSET

1,2

COMMENTS

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....

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (8,-13).

FORMULA

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

MATHEMATICA

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 *)

PROG

(MAGMA) Z<x>:= PolynomialRing(Integers()); N<r>:=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

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

(MAGMA) 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

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

CROSSREFS

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

Sequence in context: A069325 A295348 A082135 * A344055 A316594 A037697

Adjacent sequences: A153591 A153592 A153593 * A153595 A153596 A153597

KEYWORD

nonn,easy

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