

A102702


Expansion of (2x2*x^2x^3)/(1xx^2)^2.


2



2, 3, 6, 10, 18, 31, 54, 93, 160, 274, 468, 797, 1354, 2295, 3882, 6554, 11046, 18587, 31230, 52401, 87812, 146978, 245736, 410425, 684818, 1141611, 1901454, 3164458, 5262330, 8744599, 14521158, 24097797, 39965224, 66241330, 109731132
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OFFSET

0,1


COMMENTS

A floretiongenerated sequence which results from a certain transform of the Fibonacci numbers. Specifically, (a(n)) is the (type 1B) tesfortransform 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 lesfortransform (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).
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


REFERENCES

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Chapter 15, page 187, "Hosoya's Triangle".
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. E. Hoggatt, Jr. and M. BicknellJohnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117122.
Mengmeng Liu and Andrew Yezhou Wang, The number of designated parts in compositions with restricted parts, Journal of Integer Sequences, 23 (2020), Article 20.1.8.
Index entries for linear recurrences with constant coefficients, signature (2,1,2,1).


FORMULA

G.f.: (2x2*x^2x^3)/(1xx^2)^2.
a(n) = 2*F(n+1) + A001629(n+3)  2*A029907(n+1);
F(n+1) = a(n+2)  a(n+1)  a(n).
a(0)=2, a(1)=3, a(2)=6, a(3)=10, a(n)=2*a(n1)+a(n2)2*a(n3)a(n4).  Harvey P. Dale, Apr 21 2014
a(n) = A010049(n+1) + A000045(n+2).  R. J. Mathar, May 21 2019
a(n) = ((2*n+10)*F(n+1)(n4)*F(n))/5.  Andrew Yezhou Wang, Jul 14 2019


MATHEMATICA

CoefficientList[Series[(2x2x^2x^3)/(x^4+2x^3x^22x+1), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 1, 2, 1}, {2, 3, 6, 10}, 40] (* Harvey P. Dale, Apr 21 2014 *)


PROG

Floretion Algebra Multiplication Program. FAMP Code: (a(n)) = 2tesforseq[ + .5'i + .5i' ], 2lesforseq = A001629, jesforseq = A029907, vesforseq = A000045, ForType: 1B.
(Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (2x2*x^2x^3)/(1xx^2)^2)); // Marius A. Burtea, Dec 31 2019


CROSSREFS

Cf. A000045, A001629, A029907.
Sequence in context: A347787 A215006 A172516 * A181532 A077930 A060945
Adjacent sequences: A102699 A102700 A102701 * A102703 A102704 A102705


KEYWORD

easy,nonn


AUTHOR

Creighton Dement, Feb 04 2005


EXTENSIONS

Corrected by T. D. Noe, Nov 02 2006


STATUS

approved



