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 A270863 Self-composition of the Fibonacci sequence. 6
 0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027. From Oboifeng Dira, Jun 28 2020: (Start) This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where 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). Some cases of k values are: k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571 k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570 k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569 k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568 k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567 k=0, f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1) k=1, f(x) g.f. A000045 and g(x) g.f. A000045 k=2, f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1) k=3, f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n k=4, f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n k=5, f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n k=6, f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2) k=7, f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2). (End) LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853. Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683 Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1). FORMULA 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. G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016 G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016 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 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 EXAMPLE a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50. MAPLE f:= x-> x/(1-x-x^2): a:= n-> coeff(series(f(f(x)), x, n+1), x, n): seq(a(n), n=0..30); PROG (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 (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 (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 CROSSREFS Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834. Sequence in context: A244406 A244407 A173993 * A027914 A098703 A025272 Adjacent sequences: A270860 A270861 A270862 * A270864 A270865 A270866 KEYWORD nonn,easy AUTHOR Oboifeng Dira, Mar 24 2016 STATUS approved

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Last modified September 17 04:18 EDT 2024. Contains 375984 sequences. (Running on oeis4.)