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A140455
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13-Fibonacci sequence.
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8
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0, 1, 13, 170, 2223, 29069, 380120, 4970629, 64998297, 849948490, 11114328667, 145336221161, 1900485203760, 24851643870041, 324971855514293, 4249485765555850, 55568286807740343, 726637214266180309
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The k-Fibonacci sequences for k=2..12 are A000129, A006190, A001076, A052918,
A005668, A054413, A041025, A099371, A041041, A049666, A041061. This here is
k=13. k=14 is A041085, k=16 A041113, k=18 A041145, k=20 A041181, k=22 A041221.
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765.
(End)
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 13's along the main diagonal and 1's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 08 2011]
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LINKS
| Sergio Falcon, Angel Plaza, The k-Fibonacci sequence and Pascal 2-triangle, Chaos, Solit. Fract. 33 (2007) 38-49.
Tanya Khovanova, Recursive sequences. [From Johannes W. Meijer, Jun 12 2010]
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FORMULA
| O.g.f.: x/(1-13*x-x^2).
a(n)=13*a(n-1)+a(n-2).
a(n-r)*a(n+r)-a(n)^2=(-1)^(n+1-r)*a(r)^2.
a(n)=sum_{i=0..floor((n-1)/2)} binomial(n,2i+1)*13^(n-1-2*i)*(13^2+4)^i/2^(n-1).
a(n)=((13+sqrt(173))^n-(13-sqrt(173))^n)/(2^n*sqrt(173)). [From Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009]
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
a(2*n) = 13*A097844(n), a(2*n+1) = A098244(n).
a(3*n+1) = A041319(5*n), a(3*n+2) = A041319(5*n+3), a(3*n+3) = 2*A041319(5*n+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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MAPLE
| F := proc(n, k) coeftayl( x/(1-k*x-x^2), x=0, n) ; end: for n from 0 to 20 do printf("%d, ", F(n, 13)) ; od:
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MATHEMATICA
| a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*13, {n, 3*4!}]; lst [From Vladimir Orlovsky, Oct 27 2009]
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PROG
| (Sage) [lucas_number1(n, 13, -1) for n in xrange(0, 18)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2009]
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CROSSREFS
| Sequence in context: A057684 A053153 A167254 * A041314 A065544 A096719
Adjacent sequences: A140452 A140453 A140454 * A140456 A140457 A140458
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KEYWORD
| easy,nonn
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AUTHOR
| R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2008
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