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%I #184 Mar 28 2023 17:12:46
%S 0,2,8,34,144,610,2584,10946,46368,196418,832040,3524578,14930352,
%T 63245986,267914296,1134903170,4807526976,20365011074,86267571272,
%U 365435296162,1548008755920,6557470319842,27777890035288,117669030460994,498454011879264,2111485077978050
%N Even Fibonacci numbers; or, Fibonacci(3*n).
%C a(n) = 3^n*b(n;2/3) = -b(n;-2), but we have 3^n*a(n;2/3) = F(3n+1) = A033887 and a(n;-2) = F(3n-1) = A015448, where a(n;d) and b(n;d), n=0,1,...,d, denote the so-called delta-Fibonacci numbers (the argument "d" of a(n;d) and b(n;d) is abbreviation of the symbol "delta") defined by the following equivalent relations: (1 + d*((sqrt(5) - 1)/2))^n = a(n;d) + b(n;d)*((sqrt(5) - 1)/2) equiv. a(0;d)=1, b(0;d)=0, a(n+1;d) = a(n;d) + d*b(n;d), b(n+1;d) = d*a(n;d) + (1-d)b(n;d) equiv. a(0;d)=a(1;d)=1, b(0;1)=0, b(1;d)=d, and x(n+2;d) + (d-2)*x(n+1;d) + (1-d-d^2)*x(n;d) = 0 for every n=0,1,...,d, and x=a,b equiv. a(n;d) = Sum_{k=0..n} C(n,k)*F(k-1)*(-d)^k, and b(n;d) = Sum_{k=0..n} C(n,k)*(-1)^(k-1)*F(k)*d^k equiv. a(n;d) = Sum_{k=0..n} C(n,k)*F(k+1)*(1-d)^(n-k)*d^k, and b(n;d) = Sum_{k=1..n} C(n;k)*F(k)*(1-d)^(n-k)*d^k. The sequences a(n;d) and b(n;d) for special values d are connected with many known sequences: A000045, A001519, A001906, A015448, A020699, A033887, A033889, A074872, A081567, A081568, A081569, A081574, A081575, A163073 (see also the papers of Witula et al.). - _Roman Witula_, Jul 12 2012
%C For any odd k, Fibonacci(k*n) = sqrt(Fibonacci((k-1)*n) * Fibonacci((k+1)*n) + Fibonacci(n)^2). - _Gary Detlefs_, Dec 28 2012
%C The ratio of consecutive terms approaches the continued fraction 4 + 1/(4 + 1/(4 +...)) = A098317. - _Hal M. Switkay_, Jul 05 2020
%D Arthur T. Benjamin and Jennifer J. Quinn,, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, id. 232.
%H T. D. Noe, <a href="/A014445/b014445.txt">Table of n, a(n) for n = 0..200</a>
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/azarianIJCMS37-40-2012.pdf">Fibonacci Identities as Binomial Sums</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38 (2012), pp. 1871-1876 (See Corollary 1 (v)).
%H H. H. Ferns, <a href="https://www.fq.math.ca/Scanned/5-2/elementary5-2.pdf">Problem B-115</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 2 (1967), p. 202; <a href="https://www.fq.math.ca/Scanned/6-1/elementary6-1.pdf">Identities for F_{kn} and L{kn}</a>, Solution to Problem B-115 by Stanley Rabinowitz, ibid., Vol. 6, No. 1 (1968), pp. 92-93.
%H Ira M. Gessel and Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq., Vol. 16 (2013), Article 13.4.5.
%H Edyta Hetmaniok, Bozena Piatek and Roman Wituła, <a href="https://doi.org/10.1515/math-2017-0047">Binomials Transformation Formulae of Scaled Fibonacci Numbers</a>, Open Math., Vol. 15 (2017), pp. 477-485.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html">Mathematics of the Fibonacci Series</a>.
%H Bahar Kuloğlu, Engin Özkan, and Marin Marin, <a href="https://doi.org/10.2478/auom-2023-0023">Fibonacci and Lucas Polynomials in n-gon</a>, An. Şt. Univ. Ovidius Constanţa (Romania 2023) Vol. 31, No 2, 127-140.
%H Peter McCalla and Asamoah Nkwanta, <a href="https://arxiv.org/abs/1901.07092">Catalan and Motzkin Integral Representations</a>, arXiv:1901.07092 [math.NT], 2019.
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H Roman Witula and Damian Slota, <a href="http://dx.doi.org/10.2298/AADM0902310W">delta-Fibonacci numbers</a>, Appl. Anal. Discr. Math., Vol. 3, No. 2 (2009), pp. 310-329, <a href="http://www.ams.org/mathscinet-getitem?mr=2555042">MR2555042</a>.
%H Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0436">Binomials transformation formulae of scaled Lucas numbers</a>, Demonstratio Math., Vol. 46 (2013), pp. 15-27.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,1).
%F G.f.: 2*x + 8*x^2 + 34*x^3 + 144*x^4 + 610*x^5 + 2584*x^6 + 10946*x^7 + ...
%F a(n) = Sum_{k=0..n} binomial(n, k)*F(k)*2^k. - _Benoit Cloitre_, Oct 25 2003
%F From _Lekraj Beedassy_, Jun 11 2004: (Start)
%F a(n) = 4*a(n-1) + a(n-2), with a(-1) = 2, a(0) = 0.
%F a(n) = 2*A001076(n).
%F a(n) = (F(n+1))^3 + (F(n))^3 - (F(n-1))^3. (End)
%F a(n) = Sum_{k=0..floor((n-1)/2)} C(n, 2*k+1)*5^k*2^(n-2*k). - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
%F a(n) = Sum_{k=0..n} F(n+k)*binomial(n, k). - _Benoit Cloitre_, May 15 2005
%F O.g.f.: 2*x/(1 - 4*x - x^2). - _R. J. Mathar_, Mar 06 2008
%F a(n) = second binomial transform of (2,4,10,20,50,100,250). This is 2* (1,2,5,10,25,50,125) or 5^n (offset 0): *2 for the odd numbers or *4 for the even. The sequences are interpolated. Also a(n) = 2*((2+sqrt(5))^n - (2-sqrt(5))^n)/sqrt(20). - Al Hakanson (hawkuu(AT)gmail.com), May 02 2009
%F a(n) = 3*F(n-1)*F(n)*F(n+1) + 2*F(n)^3, F(n)=A000045(n). - _Gary Detlefs_, Dec 23 2010
%F a(n) = (-1)^n*3*F(n) + 5*F(n)^3, n >= 0. See the D. Jennings formula given in a comment on A111125, where also the reference is given. - _Wolfdieter Lang_, Aug 31 2012
%F With L(n) a Lucas number, F(3*n) = F(n)*(L(2*n) + (-1)^n) = (L(3*n+1) + L(3*n-1))/5 starting at n=1. - _J. M. Bergot_, Oct 25 2012
%F a(n) = sqrt(Fibonacci(2*n)*Fibonacci(4*n) + Fibonacci(n)^2). - _Gary Detlefs_, Dec 28 2012
%F For n > 0, a(n) = 5*F(n-1)*F(n)*F(n+1) - 2*F(n)*(-1)^n. - _J. M. Bergot_, Dec 10 2015
%F a(n) = -(-1)^n * a(-n) for all n in Z. - _Michael Somos_, Nov 15 2018
%F a(n) = (5*Fibonacci(n)^3 + Fibonacci(n)*Lucas(n)^2)/4 (Ferns, 1967). - _Amiram Eldar_, Feb 06 2022
%F a(n) = 2*i^(n-1)*S(n-1,-4*i), with i = sqrt(-1), and the Chebyshev S-polynomials (see A049310) with S(-1, x) = 0. From the simplified trisection formula. - _Gary Detlefs_ and _Wolfdieter Lang_, Mar 04 2023
%p a:= n-> (<<0|1>, <1|1>>^(3*n))[1,2]:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 03 2023
%t Table[Fibonacci[3n], {n,0,30}] (* _Stefan Steinerberger_, Apr 07 2006 *)
%t LinearRecurrence[{4,1},{0,2},30] (* _Harvey P. Dale_, Nov 14 2021 *)
%t Table[ SeriesCoefficient[2*x/(1 - 4*x - x^2), {x, 0, n}], {n, 0, 20}] (* _Nikolaos Pantelidis_, Feb 02 2023 *)
%o (MuPAD) numlib::fibonacci(3*n) $ n = 0..30; // _Zerinvary Lajos_, May 09 2008
%o (Sage) [fibonacci(3*n) for n in range(0, 30)] # _Zerinvary Lajos_, May 15 2009
%o (Magma) [Fibonacci(3*n): n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011
%o (PARI) a(n)=fibonacci(3*n) \\ _Charles R Greathouse IV_, Oct 25 2012
%Y Cf. A000032, A000045, A001076.
%Y First differences of A099919.
%Y Third column of array A102310.
%K nonn,easy,nice
%O 0,2
%A _Mohammad K. Azarian_
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