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A081568 Third binomial transform of Fibonacci(n+1). 7

%I

%S 1,4,17,75,338,1541,7069,32532,149965,691903,3193706,14745009,

%T 68084297,314394980,1451837593,6704518371,30961415074,142980203437,

%U 660285858245,3049218769908,14081386948661,65028302171639,300302858766202

%N Third binomial transform of Fibonacci(n+1).

%C Binomial transform of A081567.

%C Case k=3 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.

%C a(n) = 4^n*a(n;1/4) = Sum_{k=0..n} binomial(n,k)(-1)^k F(k-1) 4^(n-k), which also implies Deléham's formula given below and where a(n;d), n=0,1,..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - _Roman Witula_, Jul 12 2012

%D D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012).

%H Vincenzo Librandi, <a href="/A081568/b081568.txt">Table of n, a(n) for n = 0..200</a>

%H R. Witula, Damian Slota, <a href="http://dx.doi.org/10.2298/AADM0902310W">delta-Fibonacci numbers</a>, Appl. Anal. Discr. Math 3 (2009) 310-329, <a href="http://www.ams.org/mathscinet-getitem?mr=2555042">MR2555042</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11).

%F a(n) = 7*a(n-1) - 11*a(n-2), a(0)=1, a(1)=4.

%F a(n) = (1/2 - sqrt(5)/10)*(7/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 7/2)^n = A099453(n)-3*A099453(n-1).

%F G.f.: (1-3*x)/(1-7*x+11*x^2).

%F a(n) = Sum_{k=0..n} A094441(n,k)*3^k. - _Philippe Deléham_, Dec 14 2009

%F G.f.: Q(0,u)/x -1/x, where u=x/(1-3*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013

%t CoefficientList[Series[(1 - 3 x) / (1 - 7 x + 11 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 09 2013 *)

%t LinearRecurrence[{7,-11},{1,4},30] (* _Harvey P. Dale_, Feb 01 2015 *)

%o (MAGMA) I:=[1, 4]; [n le 2 select I[n] else 7*Self(n-1)-11*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 09 2013

%o (PARI) Vec((1-3*x)/(1-7*x+11*x^2) + O(x^100)) \\ _Altug Alkan_, Dec 10 2015

%Y Cf. A000045, A161731 (INVERT transform), A007582 (INVERTi transform), A081569 (binomial transform).

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 22 2003

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Last modified March 23 11:07 EDT 2019. Contains 321424 sequences. (Running on oeis4.)