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A162268
a(n) = ((5+sqrt(2))*(1+sqrt(2))^n + (5-sqrt(2))*(1-sqrt(2))^n)/2.
2
5, 7, 19, 45, 109, 263, 635, 1533, 3701, 8935, 21571, 52077, 125725, 303527, 732779, 1769085, 4270949, 10310983, 24892915, 60096813, 145086541, 350269895, 845626331, 2041522557, 4928671445, 11898865447, 28726402339, 69351670125
OFFSET
0,1
COMMENTS
Binomial transform of A162396.
FORMULA
a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 5, a(1) = 7.
G.f.: (5-3*x)/(1-2*x-x^2).
a(n) = 5*A000129(n+1) - 3*A000129(n). - R. J. Mathar, Mar 06 2013
a(n) = 4*A001333(n) + A001333(n+1). - G. C. Greubel, Aug 17 2018
MATHEMATICA
LinearRecurrence[{2, 1}, {5, 7}, 30] (* Vincenzo Librandi, Feb 03 2018 *)
Table[(4*LucasL[n, 2] + LucasL[n + 1, 2])/2, {n, 0, 30}] (* G. C. Greubel, Aug 17 2018 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+r)*(1+r)^n+(5-r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
(PARI) x='x+O('x^30); Vec((5-3*x)/(1-2*x-x^2)) \\ G. C. Greubel, Aug 17 2018
CROSSREFS
Cf. A162396.
Sequence in context: A036061 A288608 A079361 * A045447 A337439 A159048
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009
EXTENSIONS
Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009
STATUS
approved