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%I #51 Sep 08 2022 08:45:09
%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) for n >= 2, with a(0) = 1 and 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
%H Vincenzo Librandi, <a href="/A081568/b081568.txt">Table of n, a(n) for n = 0..200</a>
%H Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, <a href="https://doi.org/10.1515/math-2017-0047">Binomials Transformation Formulae of Scaled Fibonacci Numbers</a>, Open Mathematics, 15(1) (2017), 477-485.
%H Roman Witula and 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) for n >= 2, with a(0) = 1 and 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
%p seq(coeff(series((1-3*x)/(1-7*x+11*x^2), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 12 2019
%t CoefficientList[Series[(1-3x)/(1 -7x +11x^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^30)) \\ _Altug Alkan_, Dec 10 2015
%o (Sage)
%o def A081568_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-3*x)/(1-7*x+11*x^2)).list()
%o A081568_list(30) # _G. C. Greubel_, Aug 12 2019
%o (GAP) a:=[1,4];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # _G. C. Greubel_, Aug 12 2019
%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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