A003410 Expansion of (1+x)(1+x^2)/(1-x-x^3).

1, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452, 9651449

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Emeric Deutsch, Feb 15 2010: (Start)

a(n) is the number of binary words of length n that have no pair of adjacent 1's and have no 0000 subwords. Example: a(4)=7 because we have 0101, 1010, 0010, 1001, 0100, 0001, and 1000.

a(n) = A171855(n,0). (End)

a(n) is the number of solus bitstrings of length n with no runs of 4 zeros. - Steven Finch, Mar 25 2020

References

• R. K. Guy, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
• R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Crossrefs

Essentially the same as A058278 and A097333, partial sums and first differences of A058278, first and second differences of itself and A038718. Equals A038718(n+1) + 1, n>0.

Cf. A171855. - Emeric Deutsch, Feb 15 2010

Programs

Maple

G:=series((1+x)*(1+x^2)/(1-x-x^3),x=0,42): 1,seq(coeff(G,x^n),n=1..38);
A003410:=-(1+z)*(1+z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation

Mathematica

Join[{1}, LinearRecurrence[{1, 0, 1}, {2, 3, 5}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[((1+x)(1+x^2))/(1-x-x^3),{x,0,50}],x] (* Harvey P. Dale, Jul 07 2025 *)

PARI

a(n)=([0,1,0; 0,0,1; 1,0,1]^n*[1;2;3])[1,1] \\ Charles R Greathouse IV, Mar 25 2020

Formula

a(n) = a(n-1) + a(n-3) for n>3, see also A000930. - Reinhard Zumkeller, Oct 26 2005

For n>1, a(n) = 2*A000930(n) + A000930(n-2). - Gerald McGarvey, Sep 10 2008

a(n) = A058278(n+4) = A097333(n+1) for n >= 1. - Jianing Song, Aug 11 2023

Extensions

More terms from Emeric Deutsch, Dec 11 2004