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Search: id:A000211
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| A000211 |
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a(n) = a(n-1) + a(n-2) - 2.
(Formerly M2396 N0953)
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+0
3
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| 4, 3, 5, 6, 9, 13, 20, 31, 49, 78, 125, 201, 324, 523, 845, 1366, 2209, 3573, 5780, 9351, 15129, 24478, 39605, 64081, 103684, 167763, 271445, 439206, 710649, 1149853, 1860500, 3010351, 4870849, 7881198
(list; graph; listen)
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OFFSET
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0,1
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. Metropolis et al., Permanents of cyclic (0,1) matrices, J. Combin. Theory, 7 (1969), 291-321.
H. Minc, Permanents of (0,1)-circulants, Canad. Math. Bull., 7 (1964), 253-263.
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233.
R. P. Stanley, Enumerative Combinatorics I, Example 4.7.15, p. 252.
K. Yamamoto, Structure polynomial of Latin rectangles and its application to a combinatorial problem, Memoirs of the Faculty of Science, Kyusyu University, Series A, 10 (1956), 1-13.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..500
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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G.f.: 2*x/(1-x)+x*(1+2*x)/(1-x-x^2). a(n) = Lucas number A000032(n) + 2.
Binomial transform of [4, -1, 3, -4, 7, -11, 18,...], i.e. the series continues as a signed version of the Lucas series, A000204. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 08 2007
a(n)=F(n)+F(n+2)+2, n>=-1 {where F(n) is the n-th Fibonacci number} - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
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MAPLE
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A000211:=-(1+z)*(4*z-3)/(z-1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for the leading 4.]
with(combinat): seq(fibonacci(n)+fibonacci(n+2)+2, n=-1..32); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
(Maple) a := n -> (Matrix([[4, 1, 5]]). Matrix(3, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 0, -1][i] else 0 fi)^n)[1, 1] ; seq (a(n), n=0..33); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 01 2008]
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CROSSREFS
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Cf. A000204.
Sequence in context: A133981 A016701 A023829 this_sequence A059902 A068982 A035427
Adjacent sequences: A000208 A000209 A000210 this_sequence A000212 A000213 A000214
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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