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A045873 A006496/2. 12
0, 1, 2, -1, -12, -19, 22, 139, 168, -359, -1558, -1321, 5148, 16901, 8062, -68381, -177072, -12239, 860882, 1782959, -738492, -10391779, -17091098, 17776699, 121008888, 153134281, -298775878, -1363223161, -1232566932 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partial sums of A006495. - Paul Barry, Mar 16 2006

This is the Lucas U(P=2,Q=5) sequence. - R. J. Mathar, Oct 24 2012

With different signs, 0, 1, -2, -1, 12, -19, -22, 139, -168, -359, 1558,.. we obtain the Lucas U(-2,5) sequence. - R. J. Mathar, Jan 08 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (2,-5).

Index entries for Lucas sequences

FORMULA

Contribution from Paul Barry, Sep 20 2003: (Start)

G.f.: x/(1-2x+5x^2);

E.g.f.: exp(x)sin(2x)/2;

a(n) = 2*a(n-1)-5*a(n-2), a(0)=0, a(1)=1;

a(n) = ((1+2i)^n-(1-2i)^n)/(4i), where i=sqrt(-1);

a(n) = Im{(1+2i)^n/2};

a(n) = sum{k=0..floor(n/2), C(n, 2k+1)(-4)^k}. (End)

a(n+1) = sum{k=0..n, C(k,n-k)2^k*(-5/2)^(n-k)}. - Paul Barry, Mar 16 2006

G.f.: 1/(4*x - 1/G(0)) where G(k) =  1 - (k+1)/(1 - x/(x - (k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 5*x)/( x*(4*k+4 - 5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013

MATHEMATICA

Join[{a=0, b=1}, Table[c=2*b-5*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)

PROG

(Sage) [lucas_number1(n, 2, 5) for n in xrange(0, 29)] # [From Zerinvary Lajos, Apr 23 2009]

(PARI) concat(0, Vec(1/(1-2*x+5*x^2)+O(x^99))) \\ Charles R Greathouse IV, Dec 22 2011

CROSSREFS

Cf. A084102, A088136, A088137, A088139.

a(n)^2 = A094423(n).

Sequence in context: A151508 A164826 A055392 * A265022 A110060 A061081

Adjacent sequences:  A045870 A045871 A045872 * A045874 A045875 A045876

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Paul Barry, Sep 20 2003

STATUS

approved

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Last modified December 10 11:39 EST 2016. Contains 279001 sequences.