A102702 Expansion of (2-x-2*x^2-x^3)/(1-x-x^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

Offset

0,1

Comments

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).

Floretion Algebra Multiplication Program. FAMP Code: (a(n)) = 2tesforseq[ + .5'i + .5i' ], 2lesforseq = A001629, jesforseq = A029907, vesforseq = A000045, ForType: 1B.

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. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.

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.: (2-x-2*x^2-x^3)/(1-x-x^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(n-1)+a(n-2)-2*a(n-3)-a(n-4). - 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)-(n-4)*F(n))/5. - Andrew Yezhou Wang, Jul 14 2019

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000045, A001629, A029907.

Sequence in context: A215006 A364463 A172516 * A077930 A181532 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