A077922 -id:A077922 - cp's OEIS frontend

A077939 Expansion of 1/(1 - 2*x - x^2 - x^3).

1, 2, 5, 13, 33, 84, 214, 545, 1388, 3535, 9003, 22929, 58396, 148724, 378773, 964666, 2456829, 6257097, 15935689, 40585304, 103363394, 263247781, 670444260, 1707499695, 4348691431, 11075326817, 28206844760, 71837707768, 182957587113, 465959726754

Offset: 0

N. J. A. Sloane, Nov 17 2002

These coefficients are called the Tripell numbers by Bravo et al. - Rigoberto Florez, Jan 23 2020

Michael De Vlieger, Table of n, a(n) for n = 0..2463
Jhon Jairo Bravo, Maribel Diaz, and José Luis Ramirez, The 2-adic and 3-adic valuation of the Tripell sequence and an application, Turk J Math, (2020) 44: 131-141.
Jhon J. Bravo, Jose L. Herrera, and José L. Ramírez, Combinatorial Interpretation of Generalized Pell Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.2.1.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (2,1,1).

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

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

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

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 *)
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 *)

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 */

Vec(1/(1-2*x-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

def A077939_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return (1/(1-2*x-x^2-x^3)).list()
A077939_list(30) # G. C. Greubel, Feb 05 2020

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
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
G.f. for sequence with 1 prepended: 1/(1 - Sum_{k>=0} x*(x+x^2+x^3)^k). - Joerg Arndt, Sep 30 2012
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
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

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A010907 Pisot sequence E(4,19), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).

Original entry on oeis.org

4, 19, 90, 426, 2016, 9541, 45154, 213697, 1011348, 4786332, 22651920, 107203069, 507352048, 2401107571, 11363544486, 53779407822, 254517831936, 1204537747753, 5700626846950, 26978935702753, 127681216679304, 604267465267128, 2859772009358880, 13534231802298265, 64052459384483260, 303136344428812723, 1434630991482656082, 6789572149788327282

Offset: 0

Simon Plouffe

nonn

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)
Index entries for linear recurrences with constant coefficients, signature (4, 3, 2, 1).

Cf. A077922.

PSE[a_,b_,n_]:=Join[{x=a,y=b}, Table[z=Floor[y^2/x+1/2]; x=y; y=z, {n}]]; A010907=PSE[4,19,20] (* Zak Seidov, Mar 24 2011 *)
nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; Transpose[NestList[nxt,{4,19},20]] [[1]] (* Harvey P. Dale, Mar 13 2016 *)

Theorem: a(n) = 4 a(n - 1) + 3 a(n - 2) + 2 a(n - 3) + a(n - 4). (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016

G.f.: -(x^3+2*x^2+3*x+4)/(x^4+2*x^3+3*x^2+4*x-1). [Colin Barker, Nov 29 2012] (This follows from the above recurrence. - N. J. A. Sloane, Sep 09 2016)

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A077939 Expansion of 1/(1 - 2*x - x^2 - x^3).

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