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A003410 Expansion of (1+x)(1+x^2)/(1-x-x^3).
(Formerly M0648)
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
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
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
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 *)
PROG
(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
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
Sequence in context: A076972 A301756 A170877 * A362757 A018133 A261081
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Dec 11 2004
STATUS
approved

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