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A088316
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a(n) = 13a(n-1) + a(n-2).
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4
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2, 13, 171, 2236, 29239, 382343, 4999698, 65378417, 854919119, 11179326964, 146186169651, 1911599532427, 24996980091202, 326872340718053, 4274337409425891, 55893258663254636, 730886700031736159
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OFFSET
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0,1
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COMMENTS
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a(n+1)/a(n) converges to (13+sqrt(173))/2 = 13.07647321... a(0)/a(1)=2/13; a(1)/a(2)=13/171; a(2)/a(3)=171/2236; a(3)/a(4)= 2236/29239; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.07647321... = 2/(13+sqrt(173)) = (sqrt(173)-13)/2.
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413.
(End)
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LINKS
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Table of n, a(n) for n=0..16.
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
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FORMULA
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a(n) = 13a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13. a(n) = [(13+sqrt(173))/2]^n + [(13-sqrt(173))/2]^n.
G.f.: (2-13*x)/(1-13*x-x^2). [From Philippe DELEHAM, Nov 02 2008]
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 13*A097845(n).
a(3n+1) = A041318(5n), a(3n+2) = A041318(5n+3), a(3n+3) = 2*A041318(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A088316(n) + A140455(n)*sqrt(173))/2.
Limit(A088316(n)/A140455(n), n=infinity) = sqrt(173).
(End)
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EXAMPLE
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a(4) = 29239 = 13a(3) + a(2) = 13*2236 + 171 = [(13+sqrt(173))/2]^4 + [(13-sqrt(173))/2]^4 = 29238.9999657 + 0.0000342 =29239.
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CROSSREFS
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Cf. A006905.
Sequence in context: A177448 A078363 A143851 * A006905 A119400 A182314
Adjacent sequences: A088313 A088314 A088315 * A088317 A088318 A088319
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KEYWORD
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easy,nonn
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AUTHOR
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Nikolay V. Kosinov, Dmitry V. Polyakov (kosinov(AT)unitron.com.ua), Nov 06 2003
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STATUS
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approved
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