This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003410 M0648 #65 Jul 07 2025 16:32:26
%S A003410 1,2,3,5,7,10,15,22,32,47,69,101,148,217,318,466,683,1001,1467,2150,
%T A003410 3151,4618,6768,9919,14537,21305,31224,45761,67066,98290,144051,
%U A003410 211117,309407,453458,664575,973982,1427440,2092015,3065997,4493437,6585452,9651449
%N A003410 Expansion of (1+x)(1+x^2)/(1-x-x^3).
%C A003410 From _Emeric Deutsch_, Feb 15 2010: (Start)
%C A003410 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.
%C A003410 a(n) = A171855(n,0). (End)
%C A003410 a(n) is the number of solus bitstrings of length n with no runs of 4 zeros. - _Steven Finch_, Mar 25 2020
%D A003410 R. K. Guy, personal communication.
%D A003410 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003410 Vincenzo Librandi, <a href="/A003410/b003410.txt">Table of n, a(n) for n = 0..1000</a>
%H A003410 Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H A003410 R. K. Guy, <a href="/A003410/a003410.pdf">Letter to N. J. A. Sloane, Apr 1975</a>
%H A003410 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A003410 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A003410 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1).
%F A003410 a(n) = a(n-1) + a(n-3) for n>3, see also A000930. - _Reinhard Zumkeller_, Oct 26 2005
%F A003410 For n>1, a(n) = 2*A000930(n) + A000930(n-2). - _Gerald McGarvey_, Sep 10 2008
%F A003410 a(n) = A058278(n+4) = A097333(n+1) for n >= 1. - _Jianing Song_, Aug 11 2023
%p A003410 G:=series((1+x)*(1+x^2)/(1-x-x^3),x=0,42): 1,seq(coeff(G,x^n),n=1..38);
%p A003410 A003410:=-(1+z)*(1+z**2)/(-1+z+z**3); # _Simon Plouffe_ in his 1992 dissertation
%t A003410 Join[{1}, LinearRecurrence[{1, 0, 1}, {2, 3, 5}, 80]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 11 2012 *)
%t A003410 CoefficientList[Series[((1+x)(1+x^2))/(1-x-x^3),{x,0,50}],x] (* _Harvey P. Dale_, Jul 07 2025 *)
%o A003410 (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
%Y A003410 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.
%Y A003410 Cf. A171855. - _Emeric Deutsch_, Feb 15 2010
%K A003410 nonn,easy
%O A003410 0,2
%A A003410 _N. J. A. Sloane_
%E A003410 More terms from _Emeric Deutsch_, Dec 11 2004