OFFSET
0,1
COMMENTS
Literally the same as A013655(n+1), since A001060(-1) = A013655(0) = 3. - Eric W. Weisstein, Jun 30 2017
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
From Wajdi Maaloul, Jun 10 2022: (Start)
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)
REFERENCES
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alfred Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, Fib. Quart., 3 (1965), 129-130.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 52.
Paul Coleman, An Introduction to the Music of Sofia Gubaidulina
Tanya Khovanova, Recursive Sequences
Casey Mongoven, Fibonacci Pitch Sets. - From Ian Stewart, Jun 07 2012
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Pan Graph
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
a(n) = 2*Fibonacci(n) + Fibonacci(n+3). - Zerinvary Lajos, Oct 05 2007
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
a(n) = 6*F(n) + F(n-3), for F(n)=A000045. - J. M. Bergot, Jul 14 2017
MAPLE
with(combinat): a:= n-> 2*fibonacci(n)+fibonacci(n+3): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007
A001060:=-(2+3*z)/(-1+z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[Fibonacci[n+4] -Fibonacci[n-1], {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
LinearRecurrence[{1, 1}, {2, 5}, 50] (* Vincenzo Librandi, Jan 16 2012 *)
Table[Fibonacci[n+2] + LucasL[n+1], {n, 0, 40}] (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(2+3x)/(1-x-x^2), {x, 0, 40}], x] (* Eric W. Weisstein, Sep 22 2017 *)
PROG
(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
(PARI) a(n)=6*fibonacci(n)+fibonacci(n-3) \\ Charles R Greathouse IV, Jul 14 2017
(PARI) a(n)=([0, 1; 1, 1]^n*[2; 5])[1, 1] \\ Charles R Greathouse IV, Jul 14 2017
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, May 04 2000
STATUS
approved