A052924 Expansion of g.f.: (1-x)/(1 - 3*x - x^2).

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

Offset: 0

Author

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

Keywords

• easy
• nonn

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

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->infinity} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 31 2010

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Sergio Falcón, The k-Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.
• Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 909
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (3,1).

Crossrefs

A108300 (first differences), A006190 (partial sums), A355981 (primes).

Cf. A000032, A001077, A078343, A164581, A179237, A180148, A329723.

Programs

GAP

a:=[1,2];; for n in [3..30] do a[n]:=3*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Jun 09 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2) )); // G. C. Greubel, Jun 09 2019

Maple

spec:= [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Prod(Z,Z))))}, unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
seq(coeff(series((1-x)/(1-3*x-x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

Mathematica

CoefficientList[Series[(1-x)/(1-3*x-x^2), {x,0,30}], x] (* G. C. Greubel, Jun 09 2019 *)

PARI

Vec((1-x)/(1-3*x-x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011

Sage

((1-x)/(1-3*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 09 2019

Formula

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

a(n) = Sum_{alpha=RootOf(-1+3*x+x^2)} (1/13)*(1+5*alpha)*alpha^(-1-n).

With offset 1: a(1)=1; for n > 1, a(n) = Sum_{i=1..3*n-4} a(ceiling(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

From Creighton Dement, Nov 04 2004: (Start)

a(n) = A006190(n+1) - A006190(n);

4*a(n) = 9*A006190(n+1) - A006497(n+1) - 2*A003688(n+1). (End)

Numerators in continued fraction [1, 2, 3, 3, 3, ...], where the latter = 0.69722436226...; 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

a(n) = Sum_{k=0..n-2} A168561(n-2,k)*3^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024

From Peter Bala, Jul 08 2025: (Start)

The following series telescope:

Sum_{n >= 1} 1/(a(n) + 3*(-1)^(n+1)/a(n)) = 1/2, since 1/(a(n) + 3*(-1)^(n+1)/a(n)) = b(n) - b(n+1), where b(n) = (1/3) * (a(n) + a(n-1)) / (a(n)*a(n-1)).

Sum_{n >= 1} (-1)^(n+1)/(a(n) + 3*(-1)^(n+1)/a(n)) = 1/6, since 1/(a(n) + 3*(-1)^(n+1)/a(n)) = c(n) + c(n+1), where c(n) = (1/3) * (a(n) - a(n-1)) / (a(n)*a(n-1)). (End)

Extensions

More terms from James Sellers, Jun 06 2000