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A001060
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a(n) = a(n-1) + a(n-2) with a(0)=2, a(1)=5. Sometimes called the Evangelist Sequence.
(Formerly M1338 N0512)
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19
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2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, 16114, 26073, 42187, 68260, 110447, 178707, 289154, 467861, 757015, 1224876, 1981891, 3206767, 5188658, 8395425, 13584083, 21979508, 35563591, 57543099, 93106690, 150649789
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OFFSET
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0,1
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COMMENTS
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Used by the Sofia Gubaidulina and other composers. - Ian Stewart, Jun 07 2012
From a(2) on, sums of five consecutive Fibonacci numbers; the subset of primes is essentially in A153892. - R. J. Mathar, Mar 24 2010
Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (is this A001175?). - R. J. Mathar, Aug 10 2012
Also the number of independent vertex sets and vertex covers in the (n+1)-pan graph. - Eric W. Weisstein, Jun 30 2017
For n > 0, a(n) is the number of ways to tile the figure below with squares and dominoes (a strip of length n+1 that contains a vertical strip of height 3 in its second tile). For instance, a(4) is the number of ways to tile this figure (of length 5) with squares and dominoes.
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(End)
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REFERENCES
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R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Eric Weisstein's World of Mathematics, Pan Graph
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FORMULA
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a(n) = Fibonacci(n+4) - Fibonacci(n-1) for n >= 1. - Ian Stewart, Jun 07 2012
a(n) = Fibonacci(n) + 2*Fibonacci(n+2) = 5*Fibonacci(n) + 2*Fibonacci(n-1). The ratio r(n) := a(n+2)/a(n) satisfies the recurrence r(n+1) = (2*r(n) - 1)/(r(n) - 1). If M denotes the 2 X 2 matrix [2, -1; 1, -1] then [a(n+2), a(n)] = M^n[2, -1]. - Peter Bala, Dec 06 2013
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MAPLE
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with(combinat): a:= n-> 2*fibonacci(n)+fibonacci(n+3): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007
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MATHEMATICA
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CoefficientList[Series[(2+3x)/(1-x-x^2), {x, 0, 40}], x] (* Eric W. Weisstein, Sep 22 2017 *)
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PROG
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(Magma) I:=[2, 5]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jan 16 2012
(Magma) a0:=2; a1:=5; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..35]]; // Bruno Berselli, Feb 12 2013
(Sage) f=fibonacci; [f(n+4) - f(n-1) for n in (0..40)] # G. C. Greubel, Sep 19 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) - F(n-1) ); # G. C. Greubel, Sep 19 2019
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CROSSREFS
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Apart from initial term, same as A013655.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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