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%I #46 Mar 09 2024 14:56:50
%S 2,3,6,10,18,31,54,93,160,274,468,797,1354,2295,3882,6554,11046,18587,
%T 31230,52401,87812,146978,245736,410425,684818,1141611,1901454,
%U 3164458,5262330,8744599,14521158,24097797,39965224,66241330,109731132
%N Expansion of (2-x-2*x^2-x^3)/(1-x-x^2)^2.
%C A floretion-generated sequence which results from a certain transform of the Fibonacci numbers. Specifically, (a(n)) is the (type 1B) tesfor-transform of the Fibonacci numbers (A000045) with respect to the floretion + .5'i + .5i' Note, for example, that the sequence A001629, appearing in the formula given, has the name "Fibonacci numbers convolved with themselves" and that this sequence arises in FAMP (see program code) under the name: the lesfor-transform (type 1B) of the Fibonacci numbers (A000045) with respect to the floretion + .5'i + .5i' . The denominator of the generating function has roots at the golden ratio phi and -(1+phi).
%C Floretion Algebra Multiplication Program. FAMP Code: (a(n)) = 2tesforseq[ + .5'i + .5i' ], 2lesforseq = A001629, jesforseq = A029907, vesforseq = A000045, ForType: 1B.
%C a(n) is the total number of parts not greater than 2 among all compositions of n+3 in which only the last part may be equal to 1. - _Andrew Yezhou Wang_, Jul 14 2019
%D Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Chapter 15, page 187, "Hosoya's Triangle".
%D S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).
%H Vincenzo Librandi, <a href="/A102702/b102702.txt">Table of n, a(n) for n = 0..1000</a>
%H V. E. Hoggatt, Jr. and M. Bicknell-Johnson, <a href="http://www.fq.math.ca/Scanned/15-2/hoggatt1.pdf">Fibonacci convolution sequences</a>, Fib. Quart., 15 (1977), 117-122.
%H Mengmeng Liu and Andrew Yezhou Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Wang/wang61.html">The number of designated parts in compositions with restricted parts</a>, Journal of Integer Sequences, 23 (2020), Article 20.1.8.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F G.f.: (2-x-2*x^2-x^3)/(1-x-x^2)^2.
%F a(n) = 2*F(n+1) + A001629(n+3) - 2*A029907(n+1);
%F F(n+1) = a(n+2) - a(n+1) - a(n).
%F a(0)=2, a(1)=3, a(2)=6, a(3)=10, a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - _Harvey P. Dale_, Apr 21 2014
%F a(n) = A010049(n+1) + A000045(n+2). - _R. J. Mathar_, May 21 2019
%F a(n) = ((2*n+10)*F(n+1)-(n-4)*F(n))/5. - _Andrew Yezhou Wang_, Jul 14 2019
%t CoefficientList[Series[(2-x-2x^2-x^3)/(x^4+2x^3-x^2-2x+1),{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{2,3,6,10},40] (* _Harvey P. Dale_, Apr 21 2014 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (2-x-2*x^2-x^3)/(1-x-x^2)^2)); // _Marius A. Burtea_, Dec 31 2019
%Y Cf. A000045, A001629, A029907.
%K easy,nonn
%O 0,1
%A _Creighton Dement_, Feb 04 2005
%E Corrected by _T. D. Noe_, Nov 02 2006
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