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

%I A052924 #84 Jul 11 2025 01:25:41
%S A052924 1,2,7,23,76,251,829,2738,9043,29867,98644,325799,1076041,3553922,
%T A052924 11737807,38767343,128039836,422886851,1396700389,4612988018,
%U A052924 15235664443,50319981347,166195608484,548906806799,1812916028881
%N A052924 Expansion of g.f.: (1-x)/(1 - 3*x - x^2).
%C A052924 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
%C A052924 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
%C A052924 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
%C A052924 Equals partial sums of A108300 prefaced with a 1: (1, 1, 5, 16, 53, 175, 578, ...). - _Gary W. Adamson_, Feb 15 2012
%D A052924 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.
%H A052924 G. C. Greubel, <a href="/A052924/b052924.txt">Table of n, a(n) for n = 0..1000</a>
%H A052924 Sergio Falcón, <a href="http://dx.doi.org/10.1016/j.chaos.2016.03.038">The k-Fibonacci difference sequences</a>, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.
%H A052924 Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.09.022">On the Fibonacci k-numbers</a>, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
%H A052924 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=909">Encyclopedia of Combinatorial Structures 909</a>
%H A052924 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A052924 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).
%F A052924 a(n) = 3*a(n-1) + a(n-2).
%F A052924 a(n) = Sum_{alpha=RootOf(-1+3*x+x^2)} (1/13)*(1+5*alpha)*alpha^(-1-n).
%F A052924 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
%F A052924 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
%F A052924 From _Creighton Dement_, Nov 04 2004: (Start)
%F A052924 a(n) = A006190(n+1) - A006190(n);
%F A052924 4*a(n) = 9*A006190(n+1) - A006497(n+1) - 2*A003688(n+1). (End)
%F A052924 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
%F A052924 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
%F A052924 a(n) = A006190(n) + A003688(n). - _R. J. Mathar_, Jul 06 2012
%F A052924 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
%F A052924 From _Peter Bala_, Jul 08 2025: (Start)
%F A052924 The following series telescope:
%F A052924 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)).
%F A052924 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)
%p A052924 spec:= [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Prod(Z,Z))))}, unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
%p A052924 seq(coeff(series((1-x)/(1-3*x-x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 16 2019
%t A052924 CoefficientList[Series[(1-x)/(1-3*x-x^2), {x,0,30}], x] (* _G. C. Greubel_, Jun 09 2019 *)
%o A052924 (PARI) Vec((1-x)/(1-3*x-x^2)+O(x^30)) \\ _Charles R Greathouse IV_, Nov 20 2011
%o A052924 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2) )); // _G. C. Greubel_, Jun 09 2019
%o A052924 (Sage) ((1-x)/(1-3*x-x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 09 2019
%o A052924 (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
%Y A052924 A108300 (first differences), A006190 (partial sums), A355981 (primes).
%Y A052924 Cf. A000032, A001077, A078343, A164581, A179237, A180148, A329723.
%K A052924 easy,nonn
%O A052924 0,2
%A A052924 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052924 More terms from _James Sellers_, Jun 06 2000