A107458 - OEIS

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

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OFFSET

0,9

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).

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

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 *)

PROG

(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

(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

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

(Sage)

def A107458_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1-x^2-x^3)/((1+x)*(1-x-x^3)) ).list()