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A270863 Self-composition of the Fibonacci sequence. 7
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.

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.

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

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.

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 May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)