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A000285 a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2.
(Formerly M3246 N1309)
35
1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505, 16276621, 26336126, 42612747, 68948873, 111561620, 180510493, 292072113, 472582606 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1)=sum(P(4;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=3. These are the sums over the SW-NE diagonals in P(4;n,k), the (4,1) Pascal triangle A093561. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums in the Pascal (1,3) triangle A095660.

In general, for a Fibonacci sequence beginning with 1,b we have a(n)=(2^(-1-n)((1-sqrt(5))^n(1+sqrt(5)-2b)+(1+sqrt(5))^n (-1+sqrt(5)+2b)))/sqrt(5). In this case we have b=4. - Herbert Kociemba, Dec 18 2011

Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 5, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... - R. J. Mathar, Aug 10 2012

a(n) = number of independent vertex subsets (i.e. the Merrifield-Simmons index) of the tree obtained from the path tree P_{n-1} by attaching two pendant edges to one of its endpoints (n>=2). Example: if n=3, then we have the star tree with edges ab, ac, ad; it has 9 independent vertex susbsets: empty, a, b, c, d, bc, cd, bd, bcd.

REFERENCES

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 53.

R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 131.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.

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

T. D. Noe, Table of n, a(n) for n = 0..500

B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014.

A. Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, Fib. Quart., 3 (1965), 129-130.

Jia Huang, Hecke algebras with independent parameters, arXiv preprint arXiv:1405.1636 [math.RT], 2014.

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.

Index entries for linear recurrences with constant coefficients, signature (1,1)

FORMULA

G.f.: (1+3*x)/(1-x-x^2).

Row sums of A131775 starting (1, 4, 5, 9, 14, 23, ...). - Gary W. Adamson, Jul 14 2007

a(n) = 2*Fibonacci(n)+Fibonacci(n+2). - Zerinvary Lajos, Oct 05 2007

a(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5))+ 3/2* ((1+sqrt(5))^(n-1) -(1-sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). Offset 1. a(3)=5. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009

a(n) = 3*Fibonacci(n+2) - 2*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010

a(n) = A104449(n+1). - Michael Somos, Apr 07 2012

From Michael Somos, May 28 2014: (Start)

a(n) = A101220(3, 0, n+1).

a(n) = A109754(3, n+1).

a(k) = A090888(2, k-1), for k > 0.

a(-1 - n) = (-1)^n * A013655(n).

a(n) = Fibonacci(n) + Lucas(n+1), see Mathematica field. (End)

11*Fibonacci(n+1) = a(n+3)-a(n-2) = 3*a(n-1)+2*a(n). - Manfred Arens and Michel Marcus, Jul 14 2014

a(n) = 4*A000045(n) + A000045(n-1). - Paolo P. Lava, May 18 2015

EXAMPLE

G.f. = 1 + 4*x + 5*x^2 + 9*x^3 + 14*x^4 + 23*x^5 + 37*x^6 + 60*x^7 + ...

MAPLE

A000285:=-(1+3*z)/(-1+z+z**2); # Simon Plouffe in his 1992 dissertation

with(combinat):a:=n->2*fibonacci(n)+fibonacci(n+2): seq(a(n), n=0..34);

MATHEMATICA

a=1; lst={a}; s=6; Do[a=s-(a+1); AppendTo[lst, a]; s+=a, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

LinearRecurrence[{1, 1}, {1, 4}, 40] (* or *) Table[(3*LucasL[n]- Fibonacci[n])/2, {n, 40}] (* Harvey P. Dale, Jul 18 2011 *)

a[ n_] := Fibonacci[n] + LucasL[n + 1]; (* Michael Somos, May 28 2014 *)

PROG

(Haskell)

a000285 n = a000285_list !! n

a000285_list = 1 : 4 : zipWith (+) a000285_list (tail a000285_list)

-- Reinhard Zumkeller, Apr 28 2011

(Maxima) a[0]:1$ a[1]:4$ a[n]:=a[n-1]+a[n-2]$ makelist(a[n], n, 0, 30); /*Martin Ettl, Oct 25 2012*/

(PARI) Vec((1+3*x)/(1-x-x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2012

(MAGMA) a0:=1; a1:=4; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

CROSSREFS

Essentially the same as A104449, which only has A104449(0)=3 prefixed.

Cf. A090888, A101220, A109754, A091157 (subsequence of primes).

Cf. A013655, A131775.

Sequence in context: A243166 A120740 A274282 * A042031 A041493 A250474

Adjacent sequences:  A000282 A000283 A000284 * A000286 A000287 A000288

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from M. F. Hasler, Jan 18 2016

STATUS

approved

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Last modified June 27 06:23 EDT 2017. Contains 288777 sequences.