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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) 38

%I M3246 N1309

%S 1,4,5,9,14,23,37,60,97,157,254,411,665,1076,1741,2817,4558,7375,

%T 11933,19308,31241,50549,81790,132339,214129,346468,560597,907065,

%U 1467662,2374727,3842389,6217116,10059505,16276621,26336126,42612747,68948873,111561620,180510493,292072113,472582606

%N a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2.

%C 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.

%C 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

%C 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

%C 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.

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

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

%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 T. D. Noe, <a href="/A000285/b000285.txt">Table of n, a(n) for n = 0..500</a>

%H B. Avila and T. Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>.

%H A. 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. 53.

%H Jia Huang, <a href="http://arxiv.org/abs/1405.1636">Hecke algebras with independent parameters</a>, arXiv preprint arXiv:1405.1636 [math.RT], 2014.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1)

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

%F Row sums of A131775 starting (1, 4, 5, 9, 14, 23, ...). - _Gary W. Adamson_, Jul 14 2007

%F a(n) = 2*Fibonacci(n)+Fibonacci(n+2). - _Zerinvary Lajos_, Oct 05 2007

%F 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

%F a(n) = 3*Fibonacci(n+2) - 2*Fibonacci(n+1). - _Gary Detlefs_, Dec 21 2010

%F a(n) = A104449(n+1). - _Michael Somos_, Apr 07 2012

%F From _Michael Somos_, May 28 2014: (Start)

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

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

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

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

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

%F 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

%F a(n) = 4*A000045(n) + A000045(n-1). - _Paolo P. Lava_, May 18 2015

%F a(n) = (9*F(n) + F(n-3))/2. - _J. M. Bergot_, Jul 15 2017

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

%p A000285:=-(1+3*z)/(-1+z+z**2); # _Simon Plouffe_ in his 1992 dissertation

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

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

%t a[ n_] := Fibonacci[n] + LucasL[n + 1]; (* _Michael Somos_, May 28 2014 *)

%o a000285 n = a000285_list !! n

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

%o -- _Reinhard Zumkeller_, Apr 28 2011

%o (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 */

%o (PARI) Vec((1+3*x)/(1-x-x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Nov 20 2012

%o (MAGMA) a0:=1; a1:=4; [GeneralizedFibonacciNumber(a0,a1,n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013

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

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

%Y Cf. A013655, A131775.

%K nonn,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _M. F. Hasler_, Jan 18 2016

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Last modified January 22 05:11 EST 2019. Contains 319353 sequences. (Running on oeis4.)