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A270863 Self-composition of the Fibonacci sequence.

%I A270863 #70 Sep 08 2022 08:46:16
%S A270863 0,1,2,6,17,50,147,434,1282,3789,11200,33109,97878,289354,855413,
%T A270863 2528850,7476023,22101326,65338038,193158521,571033600,1688143881,
%U A270863 4990651642,14753839486,43616704857,128943855250,381196100507,1126928202714,3331532438042,9848993360069
%N A270863 Self-composition of the Fibonacci sequence.
%C A270863 This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
%C A270863 From _Oboifeng Dira_, Jun 28 2020: (Start)
%C A270863 This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
%C A270863 f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
%C A270863 Some cases of k values are:
%C A270863 k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
%C A270863 k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
%C A270863 k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
%C A270863 k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
%C A270863 k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
%C A270863 k=0,  f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
%C A270863 k=1,  f(x) g.f. A000045 and g(x) g.f. A000045
%C A270863 k=2,  f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
%C A270863 k=3,  f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
%C A270863 k=4,  f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
%C A270863 k=5,  f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
%C A270863 k=6,  f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
%C A270863 k=7,  f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
%C A270863 (End)
%H A270863 Colin Barker, <a href="/A270863/b270863.txt">Table of n, a(n) for n = 0..1000</a>
%H A270863 Oboifeng Dira, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201706&amp;filename=07_41(6).pdf">A Note on Composition and Recursion</a>, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
%H A270863 Oboifeng Dira, <a href="/A270863/a270863_1.pdf">Family of composition pairs g(f(x)) generating A270683</a>
%H A270863 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-3,-1).
%F A270863 a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
%F A270863 G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - _Colin Barker_, Mar 24 2016
%F A270863 G.f.: B(B(x)) where B(x) is the g.f. of A000045. - _Joerg Arndt_, Mar 25 2016
%F A270863 a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - _Vladimir Reshetnikov_, Aug 01 2019
%F A270863 0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - _Michael Somos_, Feb 05 2022
%e A270863 a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.
%p A270863 f:= x-> x/(1-x-x^2):
%p A270863 a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
%p A270863 seq(a(n), n=0..30);
%o A270863 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ _Charles R Greathouse IV_, Mar 24 2016
%o A270863 (PARI) concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ _Colin Barker_, Mar 24 2016
%o A270863 (Magma) I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // _Marius A. Burtea_, Aug 03 2019
%Y A270863 Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.
%K A270863 nonn,easy
%O A270863 0,3
%A A270863 _Oboifeng Dira_, Mar 24 2016