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%I #81 Sep 08 2022 08:44:59
%S 1,4,18,82,374,1706,7782,35498,161926,738634,3369318,15369322,
%T 70107974,319801226,1458790182,6654348458,30354161926,138462112714,
%U 631602239718,2881086973162,13142230386374,59948977985546,273460429154982,1247404189803818,5690100090709126
%N a(n+2) = 5*a(n+1) - 2*a(n), with a(0) = 1, a(1) = 4.
%C Main diagonal of the array: m(1,j)=3^(j-1), m(i,1)=1; m(i,j) = m(i-1,j) + m(i,j-1): 1 3 9 27 81 ... / 1 4 13 40 ... / 1 5 18 58 ... / 1 6 24 82 ... - _Benoit Cloitre_, Aug 05 2002
%C a(n) is also the number of 3 X n matrices of integers for which the upper-left hand corner is a 1, the rows and columns are weakly increasing, and two adjacent entries differ by at most 1. - _Richard Stanley_, Jun 06 2010
%C a(n) is the number of compositions of n when there are 4 types of 1 and 2 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%C If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, or A162909/A162910, or A071766/A229742, or A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n>=0), and the products numerator*denominator, term by term, are summed at each level n, then the resulting sequence of integers is a(n). - _Yosu Yurramendi_, May 23 2015
%C Number of 1’s in the substitution system {0 -> 110, 1 -> 11110} at step n from initial string "1" (1 -> 11110 -> 11110111101111011110110 -> ...) . - _Ilya Gutkovskiy_, Apr 10 2017
%H G. C. Greubel, <a href="/A052913/b052913.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=894">Encyclopedia of Combinatorial Structures 894</a>
%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).
%F G.f.: (1-x)/(1-5*x+2*x^2).
%F a(n) = Sum_{alpha=RootOf(1 - 5*z + 2*z^2)} (1/17)*(3+alpha)*alpha^(-1-n).
%F a(n) = ((17+3*sqrt(17))/34)*((5+sqrt(17))/2)^n + ((17-3*sqrt(17))/34)*((5-sqrt(17))/2)^n. - _N. J. A. Sloane_, Jun 03 2002
%F a(n) = A107839(n) - A107839(n-1). - _R. J. Mathar_, May 21 2015
%F a(n) = 2*A020698(n-1), n>1. - _R. J. Mathar_, Nov 23 2015
%F E.g.f.: (1/17)*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - _Stefano Spezia_, Oct 16 2019
%F a(n) = 3*A107839(n-1) + (-1)^n*A152594(n) with A107839(-1) = 0. - _Klaus Purath_, Jul 29 2020
%p spec := [S,{S=Sequence(Union(Prod(Sequence(Z),Union(Z,Z)),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p seq(coeff(series((1-x)/(1-5*x+2*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 16 2019
%t Transpose[NestList[{Last[#],5Last[#]-2First[#]}&, {1,4},20]][[1]] (* _Harvey P. Dale_, Mar 12 2011 *)
%t LinearRecurrence[{5, -2}, {1, 4}, 25] (* _Jean-François Alcover_, Jan 08 2019 *)
%o (PARI) Vec((1-x)/(1-5*x+2*x^2) + O(x^30)) \\ _Michel Marcus_, Mar 05 2015
%o (Magma) I:=[1,4]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..35]]; // _Vincenzo Librandi_, May 24 2015
%o (Sage)
%o def A052913_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-x)/(1-5*x+2*x^2)).list()
%o A052913_list(30) # _G. C. Greubel_, Oct 16 2019
%o (GAP) a:=[1,4];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # _G. C. Greubel_, Oct 16 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!((1-x)/(1-5*x+2*x^2))); // _Marius A. Burtea_, Oct 16 2019
%Y Cf. A007482 (inverse binomial transform).
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E Typo in definition corrected by _Bruno Berselli_, Jun 07 2010
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