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A153594
a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).
6
1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808
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....
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 range(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
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