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%I #33 Sep 08 2022 08:45:15
%S 7,19,50,131,343,898,2351,6155,16114,42187,110447,289154,757015,
%T 1981891,5188658,13584083,35563591,93106690,243756479,638162747,
%U 1670731762,4374032539,11451365855,29980065026,78488829223,205486422643,537970438706,1408424893475,3687304241719,9653487831682,25273159253327
%N Expansion of (7-2*x) / (1-3*x+x^2).
%C A Floretion integer sequence relating to Fibonacci numbers.
%C Inverse binomial transform of A013655; inversion of A097924.
%H Colin Barker, <a href="/A100545/b100545.txt">Table of n, a(n) for n = 0..1000</a>
%H Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, <a href="http://arxiv.org/abs/1502.03085">On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials</a>, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F a(n-1) = 4*Fibonacci(2*n) + Fibonacci(2*n-1) + Fibonacci(2*n+1).
%F a(n) + a(n+1) = A055849(n+2).
%F a(n) = 3*a(n-1) - a(n-2) with a(0)=7 and a(1)=19. - _Philippe Deléham_, Nov 16 2008
%F a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-17+7*sqrt(5)) + (3+sqrt(5))^n*(17+7*sqrt(5)))) / sqrt(5). - _Colin Barker_, Oct 14 2015
%F From _G. C. Greubel_, Jan 17 2020: (Start)
%F a(n) = Fibonacci(2*n+4) + Lucas(2*n+3).
%F E.g.f.: 2*exp(3*t/2)*(cosh(sqrt(5)*t/2) + (4/sqrt(5))*sinh(sqrt(5)*t/2)). (End)
%p F := proc(n) combinat[fibonacci](n) ; end: A100545 := proc(n) 4*F(2*(n+1)) + F(2*n+1)+F(2*n+3) ; end: for n from 0 to 30 do printf("%d,",A100545(n)) ; od ; # _R. J. Mathar_, Oct 26 2006
%t Table[Fibonacci[2*(n+2)] + LucasL[2*n+3], {n,0,30}] (* _G. C. Greubel_, Jan 17 2020 *)
%o (PARI) Vec((7-2*x)/(1-3*x+x^2) + O(x^30)) \\ _Michel Marcus_, Feb 11 2015
%o (Magma) [Fibonacci(2*n+4) +Lucas(2*n+3): n in [0..30]]; // _G. C. Greubel_, Jan 17 2020
%o (Sage) [fibonacci(2*n+4) +lucas_number2(2*n+3,1,-1) for n in (0..30)] # _G. C. Greubel_, Jan 17 2020
%o (GAP) List([0..30], n-> Fibonacci(2*n+4) +Lucas(1,-1,2*n+3)[2] ); # _G. C. Greubel_, Jan 17 2020
%Y Cf. A000032, A000045, A001906, A005248, A013655, A097924.
%K nonn,easy
%O 0,1
%A _Creighton Dement_, Dec 31 2004
%E Corrected and extended by _T. D. Noe_ and _R. J. Mathar_, Oct 26 2006
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