A107458 Expansion of g.f.: (1-x^2-x^3)/( (1+x)*(1-x-x^3) ).

1, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621

Offset: 0

Author

N. J. A. Sloane, Mar 08 2008

Keywords

• nonn
• easy

Comments

The sequence can be interpreted as the top-left entry of the n-th power of a 4 X 4 (0,1) matrix. There are 12 different choices (out of 2^16) for that (0,1) matrix. - R. J. Mathar, Mar 19 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^2_n.]
• Renata Passos Machado Vieira, and Francisco Regis Vieira Alves, Sequences of Tridovan and their identities, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 185-197. Sequence (T_n) is a subsequence of this sequence.
• Renata Passos Machado Vieira, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino, A note on the Tetrarrin sequence, Braz. Elect. J. Math. (2024).
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

Crossrefs

Cf. A001634, A013979, A078012, A135851.

Programs

GAP

a:=[1,0,0,0];; for n in [5..50] do a[n]:=a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 03 2020

Haskell

a107458 n = a107458_list !! n
a107458_list = 1 : 0 : 0 : 0 : zipWith (+) a107458_list
   (zipWith (+) (tail a107458_list) (drop 2 a107458_list))
-- Reinhard Zumkeller, Mar 23 2012

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1-x^2-x^3)/( (1+x)*(1-x-x^3))); // Marius A. Burtea, Jan 02 2020

Maple

seq(coeff(series( (1-x^2-x^3)/( (1+x)*(1-x-x^3) ), x, n+1), x, n), n = 0..50); # G. C. Greubel, Jan 03 2020

Mathematica

CoefficientList[Series[(1-x^2-x^3)/(1-x^2-x^3-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,1,1},{1,0,0,0},50] (* Harvey P. Dale, Jun 20 2011 *)

PARI

my(x='x+O('x^50)); Vec((1-x^2-x^3)/((1+x)*(1-x-x^3))) \\ G. C. Greubel, Apr 27 2017

Sage

def A107458_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2-x^3)/((1+x)*(1-x-x^3)) ).list()
A107458_list(50) # G. C. Greubel, Jan 03 2020

Formula

a(n) = a(n-2) + a(n-3) + a(n-4); a(0)=1, a(1)=0, a(2)=0, a(3)=0. - Harvey P. Dale, Jun 20 2011

a(n) + a(n-1) = A000930(n-4). - R. J. Mathar, Mar 19 2014