|
%I #103 Sep 08 2022 08:44:59
%S 1,2,4,9,21,49,114,265,616,1432,3329,7739,17991,41824,97229,226030,
%T 525456,1221537,2839729,6601569,15346786,35676949,82938844,192809420,
%U 448227521,1042002567,2422362079,5631308624,13091204281,30433357674,70748973084,164471408185
%N Expansion of (1 - x)/(1 - 3*x + 2*x^2 - x^3).
%C First differences of A095263. - _R. J. Mathar_, Nov 23 2011
%C Partial sums of A034943 starting (1, 1, 2, 5, 12, 28, 65, ...). - _Gary W. Adamson_, Feb 15 2012
%C a(n) is the number of n (decimal) digit integers x such that all digits of x are odd and all digits of 6x are even. - _Robert Israel_, Apr 17 2014
%C a(n) is the number of words of length n over the alphabet {0,1,2} that do not contain the substrings 01 or 12 and do not end in 0. - _Yiseth K. Rodríguez C._, Sep 11 2020
%H Vincenzo Librandi, <a href="/A052921/b052921.txt">Table of n, a(n) for n = 0..1000</a>
%H Sergio Falcón, <a href="https://www.rgnpublications.com/journals/index.php/cma/article/viewFile/1221/950">Binomial Transform of the Generalized k-Fibonacci Numbers</a>, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
%H I. M. Gessel and Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=905">Encyclopedia of Combinatorial Structures 905</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1).
%F G.f.: (1 - x)/(1 - 3*x + 2*x^2 - x^3).
%F a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3), with a(0)=1, a(1)=2, a(2)=4.
%F a(n) = Sum_{alpha=RootOf(-1 + 3*z - 2*z^2 + z^3)} (1/23)*(8 - 5*alpha + 7*alpha^2)*alpha^(-1-n).
%F From _Paul Barry_, Jun 21 2004: (Start)
%F Binomial transform of the Padovan sequence A000931(n+5).
%F a(n) = Sum_{k=0..n+1} C(n+k+1, n-2*k). (End)
%F a(n) = A000931(3*n + 5). - _Michael Somos_, Sep 18 2012
%F a(n) = Sum_{i=1..n+1} A000931(3*i). - _David Nacin_, Nov 03 2019
%e G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 21*x^4 + 49*x^5 + 114*x^6 + 265*x^7 + ...
%p spec := [S,{S=Sequence(Union(Z,Z,Prod(Sequence(Z),Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..29);
%p A052921 := proc(n): add(binomial(n+k+1, n-2*k),k=0..n+1) end: seq(A052921(n), n=0..29); # _Johannes W. Meijer_, Aug 16 2011
%t LinearRecurrence[{3,-2,1},{1,2,4},40] (* _Vincenzo Librandi_, Feb 14 2012 *)
%t CoefficientList[Series[(1-x)/(1-3*x+2*x^2-x^3),{x,0,30}],x] (* _Harvey P. Dale_, Nov 09 2019 *)
%o (Magma) I:=[1,2,4]; [n le 3 select I[n] else 3*Self(n-1)-2*Self(n-2) +Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 14 2012
%o (PARI) my(x='x+O('x^40)); Vec((1-x)/(1 -3*x +2*x^2 -x^3)) \\ _G. C. Greubel_, Oct 16 2019
%o (Sage)
%o def A077952_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-x)/(1 -3*x +2*x^2 -x^3)).list()
%o A077952_list(40) # _G. C. Greubel_, Oct 16 2019
%o (GAP) a:=[1,2,4];; for n in [4..40] do a[n]:=3*a[n-1]-2*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Oct 16 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x+2*x^2-x^3) )); // _Marius A. Burtea_, Oct 16 2019
%Y Cf. A000931, A034943.
%Y Cf. A097550, A137531.
%K nonn,easy
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
