|
%I #96 Sep 08 2022 08:45:07
%S 1,2,5,13,33,84,214,545,1388,3535,9003,22929,58396,148724,378773,
%T 964666,2456829,6257097,15935689,40585304,103363394,263247781,
%U 670444260,1707499695,4348691431,11075326817,28206844760,71837707768,182957587113,465959726754
%N Expansion of 1/(1 - 2*x - x^2 - x^3).
%C These coefficients are called the Tripell numbers by Bravo et al. - _Rigoberto Florez_, Jan 23 2020
%H Michael De Vlieger, <a href="/A077939/b077939.txt">Table of n, a(n) for n = 0..2463</a>
%H Jhon Jairo Bravo, Maribel Diaz, and José Luis Ramirez, <a href="https://doi.org/10.3906/mat-1907-40">The 2-adic and 3-adic valuation of the Tripell sequence and an application</a>, Turk J Math, (2020) 44: 131-141.
%H Jhon J. Bravo, Jose L. Herrera, and José L. Ramírez, <a href="https://www.emis.de/journals/JIS/VOL23/Bravo/bravo4.html">Combinatorial Interpretation of Generalized Pell Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.2.1.
%H Brian Hopkins and Stéphane Ouvry, <a href="https://arxiv.org/abs/2008.04937">Combinatorics of Multicompositions</a>, arXiv:2008.04937 [math.CO], 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,1).
%F a(n) = abs(A077986(n)) = A077849(n) - A077849(n-1) = |A077922(n)| + |A077922(n-1)| = Sum_{k=0..n} A077997(k). - _Ralf Stephan_, Feb 02 2004
%F a(n) = Sum_{m=1..n+1} Sum_(k=0..n-m+1} (Sum_{j=0..k} binomial(j,n-m-3*k+2*j+1) *binomial(k,j))*binomial(m+k-1,m-1). - _Vladimir Kruchinin_, Oct 11 2011
%F G.f. for sequence with 1 prepended: 1/(1 - Sum_{k>=0} x*(x+x^2+x^3)^k). - _Joerg Arndt_, Sep 30 2012
%F G.f.: Q(0)/2, where Q(k) = 1 + 1/(1- x*(4*k+2 + x+x^2)/(x*(4*k+4 + x+x^2) + 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 04 2013
%F a(n) = 2*a(n-1) + a(n-2) + a(n-3), where a(0) = 0, a(1)=1, a(2)=2. - _Rigoberto Florez_, Jan 23 2020
%p m:=30; S:=series(1/(1-2*x-x^2-x^3), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 05 2020
%t CoefficientList[Series[1/(1-2*x-x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,1,1},{1,2,5},40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 30 2012 *)
%t a[n_]:=a[n]=2a[n-1]+a[n-2]+a[n-3]; a[0]=0; a[1]=1; a[2]=2; Table[a[n], {n,30}] (* _Rigoberto Florez_, Jan 23 2020 *)
%o (Maxima)
%o a(n):=sum(sum((sum(binomial(j,n-m-3*k+2*j+1)*binomial(k,j),j,0,k))* binomial(m+k-1,m-1),k,0,n-m+1),m,1,n+1); /* _Vladimir Kruchinin_, Oct 11 2011 */
%o (PARI) Vec(1/(1-2*x-x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (Magma) I:=[1,2,5]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) +Self(n-3): n in [1..30]]; // _G. C. Greubel_, Feb 05 2020
%o (Sage)
%o def A077939_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return (1/(1-2*x-x^2-x^3)).list()
%o A077939_list(30) # _G. C. Greubel_, Feb 05 2020
%o (GAP) a:=[1,2,5];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Feb 05 2020
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 17 2002
%E Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021
|
