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%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
%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
%C 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(kociemba(AT)t-online.de), 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A. Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, Fib. Quart., 3 (1965), 129-130.
%D A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 53.
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%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 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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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, 2012
%H <a href="/index/Rea#recLCC">Index entries for sequences related to 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 (zerinvarylajos(AT)yahoo.com), 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. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009]
%F a(n)=3*Fibonacci(n+2)-2*Fibonacci(n+1) [From _Gary Detlefs_ Dec 21 2010]
%F a(n) = A104449(n+1). - Michael Somos, Apr 07 2012
%F a(n) = A101220(3, 0, n+1). a(n) = A109754(3, n+1). a(k) = A090888(2, k-1), for k > 0.
%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 a=1;lst={a};s=6;Do[a=s-(a+1);AppendTo[lst,a];s+=a,{n,5!}];lst [From _Vladimir Joseph Stephan Orlovsky_, Oct 27 2009]
%t LinearRecurrence[{1,1},{1,4},40] (* or *) Table[(3*LucasL[n]- Fibonacci[n])/2,{n,40}] (* From Harvey P. Dale, Jul 18 2011 *)
%o (Haskell)
%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. A131775.
%K easy,nonn,nice
%O 0,2
%A _N. J. A. Sloane_.
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