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%I M1338 N0512 #132 Sep 09 2022 23:37:38
%S 2,5,7,12,19,31,50,81,131,212,343,555,898,1453,2351,3804,6155,9959,
%T 16114,26073,42187,68260,110447,178707,289154,467861,757015,1224876,
%U 1981891,3206767,5188658,8395425,13584083,21979508,35563591,57543099,93106690,150649789
%N a(n) = a(n-1) + a(n-2) with a(0)=2, a(1)=5. Sometimes called the Evangelist Sequence.
%C Literally the same as A013655(n+1), since A001060(-1) = A013655(0) = 3. - _Eric W. Weisstein_, Jun 30 2017
%C Used by the Sofia Gubaidulina and other composers. - _Ian Stewart_, Jun 07 2012
%C From a(2) on, sums of five consecutive Fibonacci numbers; the subset of primes is essentially in A153892. - _R. J. Mathar_, Mar 24 2010
%C 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
%C Also the number of independent vertex sets and vertex covers in the (n+1)-pan graph. - _Eric W. Weisstein_, Jun 30 2017
%C From _Wajdi Maaloul_, Jun 10 2022: (Start)
%C 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.
%C _
%C |_|
%C _|_|_______
%C |_|_|_|_|_|_|
%C (End)
%D R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001060/b001060.txt">Table of n, a(n) for n = 0..1000</a>
%H Alfred Brousseau, <a href="http://www.fq.math.ca/Scanned/3-2/alfred1.pdf">Seeking the lost gold mine or exploring Fibonacci factorizations</a>, Fib. Quart., 3 (1965), 129-130.
%H Alfred Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 52.
%H Paul Coleman, <a href="http://ecmc.rochester.edu/paul/docs/gubaidulina_handout.pdf">An Introduction to the Music of Sofia Gubaidulina</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Casey Mongoven, <a href="http://www.caseymongoven.com/writing/Fibonacci_Pitch_Sequences.pdf">Fibonacci Pitch Sets</a>. - From Ian Stewart, Jun 07 2012
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PanGraph.html">Pan Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F a(n) = 2*Fibonacci(n) + Fibonacci(n+3). - _Zerinvary Lajos_, Oct 05 2007
%F a(n) = Fibonacci(n+4) - Fibonacci(n-1) for n >= 1. - _Ian Stewart_, Jun 07 2012
%F 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
%F a(n) = 6*F(n) + F(n-3), for F(n)=A000045. - _J. M. Bergot_, Jul 14 2017
%F a(n) = -(-1)^n*A000285(-2-n) = -(-1)^n*A104449(-1-n) for all n in Z. - _Michael Somos_, Oct 28 2018
%p with(combinat): a:= n-> 2*fibonacci(n)+fibonacci(n+3): seq(a(n), n=0..40); # _Zerinvary Lajos_, Oct 05 2007
%p A001060:=-(2+3*z)/(-1+z+z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[Fibonacci[n+4] -Fibonacci[n-1], {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 23 2009 *)
%t LinearRecurrence[{1,1}, {2,5}, 50] (* _Vincenzo Librandi_, Jan 16 2012 *)
%t Table[Fibonacci[n+2] + LucasL[n+1], {n, 0, 40}] (* _Eric W. Weisstein_, Jun 30 2017 *)
%t CoefficientList[Series[(2+3x)/(1-x-x^2), {x, 0, 40}], x] (* _Eric W. Weisstein_, Sep 22 2017 *)
%o (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
%o (Magma) a0:=2; a1:=5; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..35]]; // _Bruno Berselli_, Feb 12 2013
%o (PARI) a(n)=6*fibonacci(n)+fibonacci(n-3) \\ _Charles R Greathouse IV_, Jul 14 2017
%o (PARI) a(n)=([0,1; 1,1]^n*[2;5])[1,1] \\ _Charles R Greathouse IV_, Jul 14 2017
%o (Sage) f=fibonacci; [f(n+4) - f(n-1) for n in (0..40)] # _G. C. Greubel_, Sep 19 2019
%o (GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) - F(n-1) ); # _G. C. Greubel_, Sep 19 2019
%Y Cf. A000032, A000045, A000285, A104449.
%Y Apart from initial term, same as A013655.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 04 2000
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