A001060 a(n) = a(n-1) + a(n-2) with a(0)=2, a(1)=5. Sometimes called the Evangelist Sequence.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 Sequences. - _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).

Crossrefs

Cf. A000032, A000045, A000285, A104449.

Apart from initial term, same as A013655.

Programs

GAP

F:=Fibonacci;; List([0..40], n-> F(n+4) - F(n-1) ); # G. C. Greubel, Sep 19 2019

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

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 *)

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

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

a(n) = -(-1)^n*A000285(-2-n) = -(-1)^n*A104449(-1-n) for all n in Z. - Michael Somos, Oct 28 2018

E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 4*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, May 26 2025

Extensions

More terms from James Sellers, May 04 2000