A078012 Expansion of (1 - x) / (1 - x - x^3) in powers of x.

1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925

Offset

0,7

Comments

Number of compositions of n into parts >= 3. - Milan Janjic, Jun 28 2010

From Adi Dani, May 22 2011: (Start)

Number of compositions of number n into parts of the form 3*k+1, k >= 0.

For example, a(10)=19 and all compositions of 10 in parts 1,4,7 or 10 are

(1,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,4), (1,1,1,1,1,4,1), (1,1,1,1,4,1,1), (1,1,1,4,1,1,1), (1,1,4,1,1,1,1), (1,4,1,1,1,1,1), (4,1,1,1,1,1,1), (1,1,4,4), (1,4,1,4), (1,4,4,1), (4,1,1,4),(4,1,4,1), (4,4,1,1), (1,1,1,7), (1,1,7,1), (1,7,1,1), (7,1,1,1), (10). (End)

a(n+1) is for n >= 0 the number of 00's in the Narayana word NW(n); equivalently the number of two neighboring 0's at level n of the Narayana tree. See A257234. This implies that if a(0) is put to 0 then a(n) is the number of -1's in the Narayana word NW(n), and also at level n of the Narayana tree. - Wolfdieter Lang, Apr 24 2015

References

Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, page 331ff.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Christian Ballot, On Functions Expressible as Words on a Pair of Beatty Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2.

C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f_n.]

Taras Goy, On identities with multinomial coefficients for Fibonacci-Narayana sequence, Annales Mathematicae et Informaticae, Vol. 49 (2018), 75-84.

Bahar Kuloğlu, Engin Özkan, and Marin Marin, On the period of Pell-Narayana sequence in some groups, arXiv:2305.04786 [math.CO], 2023.

J. D. Opdyke, A unified approach to algorithms generating unrestricted.., J. Math. Model. Algor. 9 (2010) 53-97.

Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.

Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Formula

a(n) = Sum_{i=0..(n-3)/3} binomial(n-3-2*i, i), n >= 1, a(0) = 1.

From Michael Somos, May 03 2011: (Start)

Euler transform of A065417.

G.f.: (1 - x) / (1 - x - x^3).

a(n) = a(n-1) + a(n-3).

a(-n) = A077961(n). a(n+3) = A000930(n).

a(n+5) = A068921(n). (End)

a(n+1) = A013979(n-3) + A135851(n) + A107458(n), n >= 3.

a(n) = a(n-1) + a(n-3) for n >= 4. - Jaroslav Krizek, May 07 2011

G.f.: 1/(1 - Sum_{k>=3} x^k). - Joerg Arndt, Aug 13 2012

G.f.: Q(0)*(1-x)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013

0 = -1 + a(n)*(a(n)*(a(n) + a(n+2)) + a(n+1)*(a(n+1) - 3*a(n+2))) + a(n+1)*(+a(n+1)*(+a(n+1) + a(n+2)) + a(n+2)*(-2*a(n+2))) + a(n+2)^3 for all n in Z. - Michael Somos, Feb 03 2018

Example

G.f. = 1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 6*x^9 + 9*x^10 + 13*x^11 + ...

Maple

A078012 := proc(n): if n=0 then 1 else add(binomial(n-3-2*i, i), i=0..(n-3)/3) fi: end: seq(A078012(n), n=0..46); # Johannes W. Meijer, Aug 11 2011

Mathematica

CoefficientList[ Series[(1-x)/(1-x-x^3), {x, 0, 50}], x] (* Robert G. Wilson v, May 25 2011 *)
LinearRecurrence[{1, 0, 1}, {1, 0, 0}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
a[ n_]:= If[ n >= 0, SeriesCoefficient[ (1-x)/(1-x-x^3), {x, 0, n}], SeriesCoefficient[1/(1+x^2-x^3), {x, 0, -n}]]; (* Michael Somos, Feb 03 2018 *)

Prog

(PARI) {a(n) = if( n<0, n = -n; polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - x) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, May 03 2011 */
(Haskell)
a078012 n = a078012_list !! n
a078012_list = 1 : 0 : 0 : 1 : zipWith (+) a078012_list
   (zipWith (+) (tail a078012_list) (drop 2 a078012_list))
-- Reinhard Zumkeller, Mar 23 2012
(Magma) I:=[1, 0, 0]; [n le 3 select I[n] else Self(n-1) + Self(n-3): n in [1..50]]; // G. C. Greubel, Jan 19 2018
(Sage) ((1-x)/(1-x-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019
(GAP) a:=[1, 0, 0];; for n in [4..50] do a[n]:=a[n-1]+a[n-3]; od; a; # G. C. Greubel, Jun 28 2019

Crossrefs

Cf. A000930, A065417, A068921, A077961.

Cf. A135851, A257234.

Sequence in context: A017826 A068921 A000930 * A135851 A199804 A101913

Adjacent sequences: A078009 A078010 A078011 * A078013 A078014 A078015

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 17 2002, Mar 08 2008

Extensions

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

Status

approved