login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A052924 G.f.: (1-x)/(1-3*x-x^2). 10
1, 2, 7, 23, 76, 251, 829, 2738, 9043, 29867, 98644, 325799, 1076041, 3553922, 11737807, 38767343, 128039836, 422886851, 1396700389, 4612988018, 15235664443, 50319981347, 166195608484, 548906806799, 1812916028881 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Euler encountered this sequence when finding the largest root of z^2-3z-1=0. - V. Frederick Rickey (fred-rickey(AT)usma.edu), Aug 20 2003

Contribution from Gary W. Adamson, Jul 31 2010: (Start)

Let M = a triangle with the Pell series A000129 (1, 2, 5, 12,...) in each column,

with the leftmost column shifted upwards one row. A052924 starting

(1, 2, 7, 23,...) = Lim_{n->inf} M^n, the left-shifted vector considered

as a sequence. (End)

a(n) is the number of compositions of n when there are 2 types of 1 and 3 types of other natural numbers. - Milan Janjic, Aug 13 2010

Equals partial sums of A108300 prefaced with a 1: (1, 1, 5, 16, 53, 175, 578,...). - Gary W. Adamson, Feb 15 2012

REFERENCES

L. Euler, Introductio in analysin infinitorum, 1748, section 338. English translation by John D. Blanton, Introduction to Analysis of the Infinite, 1988, Springer, p. 286.

Sergio Falcon, The k-Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.

S. Falcon and A. Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals, 32(5) (2007), 1615-24.

LINKS

Table of n, a(n) for n=0..24.

Tanya Khovanova, Recursive Sequences

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 909

Index entries for linear recurrences with constant coefficients, signature (3,1).

FORMULA

a(n) = 3a(n-1) + a(n-2).

Sum(1/13*(5*_alpha+1)*_alpha^(-1-n), _alpha=RootOf(-1+3*_Z+_Z^2)).

With offset 1: a(1)=1, n>1 a(n)=sum(i=1, 3n-4, a(ceil(i/3))). - Benoit Cloitre, Jan 04 2004

Binomial transform of A006130. a(n)=(1/2-sqrt(13)/26)(3/2-sqrt(13)/2)^n+(1/2+sqrt(13)/26)(3/2+sqrt(13)/2)^n. - Paul Barry, Jul 20 2004

a(n) = A006190(n+1) - A006190(n); 4*a(n)_0 = 9*A006190(n+1)_0 - A006497(n+1)_0 - 2*A003688(n+1)_1 where "_" gives the offset. - Creighton Dement, Nov 04 2004

Numerators in continued fraction [1, 2, 3, 3, 3,...], where the latter = .69722436226...; the length of an inradius of a right triangle with legs 2 and 3. - Gary W. Adamson, Dec 19 2007

If p[1]=2, p[i]=3, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det A. - Milan Janjic, Apr 29 2010

a(n) = A006190(n) + A003688(n). - R. J. Mathar, Jul 06 2012

MAPLE

spec := [S, {S=Sequence(Prod(Sequence(Z), Union(Z, Z, Prod(Z, Z))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

PROG

(PARI) Vec((1-x)/(1-3*x-x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011

CROSSREFS

Cf. A108300

Sequence in context: A273345 A042575 A256030 * A067324 A091702 A274104

Adjacent sequences:  A052921 A052922 A052923 * A052925 A052926 A052927

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 25 15:14 EDT 2017. Contains 284082 sequences.