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A022112 Fibonacci sequence beginning 2, 6. 17
2, 6, 8, 14, 22, 36, 58, 94, 152, 246, 398, 644, 1042, 1686, 2728, 4414, 7142, 11556, 18698, 30254, 48952, 79206, 128158, 207364, 335522, 542886, 878408, 1421294, 2299702, 3720996, 6020698, 9741694, 15762392, 25504086, 41266478, 66770564, 108037042 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
For n>=3 the number of perfect matchings in the n-antiprism graph. - Andrew Howroyd, May 17 2017
LINKS
Tanya Khovanova, Recursive Sequences
Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee and Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
FORMULA
a(n) = 4*Fibonacci(n+2) - 2*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010
a(n) = 2*A000204(n+1). - R. J. Mathar, Mar 11 2011
G.f.: ( -2-4*x ) / ( -1+x+x^2 ). - R. J. Mathar, Mar 11 2011
a(n) = Fibonacci(n-2) + Fibonacci(n+4). - Gary Detlefs, Mar 31 2012
a(n) = L(n - 1) + L(n) + L(n + 1), for n > 0, where L(n) is the n-th Lucas number (A000204). - Alonso del Arte, Sep 25 2013
a(n) = L(n + 3) - L(n). - Bruno Berselli, Jun 15 2017
From Colin Barker, Oct 27 2017: (Start)
a(n) = (2^(-n)*((1-sqrt(5))^n*(-5+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5)))) / sqrt(5).
a(n) = a(n-1) + a(n-2) for > 1.
(End)
MATHEMATICA
LinearRecurrence[{1, 1}, {2, 6}, 40] (* Harvey P. Dale, Apr 21 2012 *)
2 LucasL[Range[30]] (* Alonso del Arte, Sep 25 2013 *)
PROG
(Haskell)
a022112 n = a022112_list !! n
a022112_list = 2 : 6 : zipWith (+) (tail a022112_list) a022112_list
-- Reinhard Zumkeller, Apr 08 2012
(PARI) a(n)=4*fibonacci(n+2)-2*fibonacci(n+1) \\ Charles R Greathouse IV, Oct 07 2015
(PARI) Vec(2*(1 + 2*x) / (1 - x - x^2) + O(x^60)) \\ Colin Barker, Oct 27 2017
CROSSREFS
Cf. sequences with formula Fibonacci(n+k)+Fibonacci(n-k) listed in A280154.
Sequence in context: A117588 A174229 A187217 * A107019 A269831 A048133
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)