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A072263 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=2, a(1)=3. 8
2, 3, 19, 72, 311, 1293, 5434, 22767, 95471, 400248, 1678099, 7035537, 29497106, 123669003, 518492539, 2173822632, 9113930591, 38210904933, 160202367754, 671661627927, 2815996722551, 11806298307288, 49498878534619, 207528127140297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Inverse binomial transform of A087130. - Johannes W. Meijer, Aug 01 2010

Pisano period lengths: 1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12... - R. J. Mathar, Aug 10 2012

This is the Lucas sequence V(3,-5). - Bruno Berselli, Jan 09 2013

LINKS

Table of n, a(n) for n=0..23.

Wikipedia, Lucas sequence: Specific names.

Index entries for linear recurrences with constant coefficients, signature (3,5).

FORMULA

a(n) = 2*A015523(n+1)-3*A015523(n).

a(n) = ((3+sqrt(29))/2)^n + ((3-sqrt(29))/2)^n.

G.f.: (2-3*x)/(1-3*x-5*x^2). - R. J. Mathar, Feb 06 2010

From Johannes W. Meijer, Aug 01 2010: (Start)

Limit(a(n+k)/a(k), k=infinity) = (A072263(n)+A015523(n)*sqrt(29))/2

Limit(A072263(n)/A015523(n)) = sqrt(29). (End)

G.f.: G(0), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 29*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

EXAMPLE

a(5)=5*b(4)+b(6): 1293=5*57+1008.

PROG

(Sage) [lucas_number2(n, 3, -5) for n in xrange(0, 16)] # Zerinvary Lajos, Apr 30 2009

CROSSREFS

Cf. A072264, A152187, A197189.

Appears in A179606 and A015523. - Johannes W. Meijer, Aug 01 2010

Sequence in context: A153409 A143893 A262957 * A009178 A141508 A119344

Adjacent sequences:  A072260 A072261 A072262 * A072264 A072265 A072266

KEYWORD

nonn,easy

AUTHOR

Miklos Kristof, Jul 08 2002

EXTENSIONS

Offset changed and terms added by Johannes W. Meijer, Jul 19 2010

STATUS

approved

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Last modified December 2 15:01 EST 2016. Contains 278678 sequences.