cp's OEIS Frontend

A005253 Number of binary words of length n in which the ones occur only in blocks of length at least 4.

%I A005253 M1044 #65 Jan 21 2025 23:56:25
%S A005253 1,1,1,1,2,4,7,11,16,23,34,52,81,126,194,296,450,685,1046,1601,2452,
%T A005253 3753,5739,8771,13404,20489,31327,47904,73252,112004,171245,261813,
%U A005253 400285,612009,935737,1430710,2187496,3344567,5113647,7818464,11953991,18277014,27944604
%N A005253 Number of binary words of length n in which the ones occur only in blocks of length at least 4.
%D A005253 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005253 R. Austin and R. K. Guy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/16-1/austin.pdf">Binary sequences without isolated ones</a>, Fib. Quart., 16 (1978), 84-86.
%H A005253 Russ Chamberlain, Sam Ginsburg and Chi Zhang, <a href="http://digital.library.wisc.edu/1793/61870">Generating Functions and Wilf-equivalence on Theta_k-embeddings</a>, University of Wisconsin, April 2012.
%H A005253 R. K. Guy, <a href="/A005251/a005251_1.pdf">Anyone for Twopins?</a>, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
%H A005253 V. C. Harris and C. C. Styles, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,3,2).
%H A005253 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%H A005253 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 A005253 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005253 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=425">Encyclopedia of Combinatorial Structures 425</a>
%H A005253 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1).
%F A005253 G.f.: (1-x+x^4)/(1-2x+x^2-x^5). - _Simon Plouffe_ in his 1992 dissertation.
%F A005253 a(n-1) = Sum_{k=0..floor(n/5)} binomial(n-3k, 2k). - _Paul Barry_, Sep 16 2004
%e A005253 a(6)=7 because 7 binary words of length 6 in which the ones occur only in blocks of length at least 4: 000000, 001111, 011110, 011111, 111100, 111110, 111111. - _Jinyuan Wang_, Jan 20 2025
%t A005253 LinearRecurrence[{2,-1,0,0,1},{1,1,1,1,2},50] (* _Harvey P. Dale_, Mar 14 2018 *)
%K A005253 nonn,easy
%O A005253 0,5
%A A005253 _N. J. A. Sloane_
%E A005253 More terms from _Harvey P. Dale_, Mar 14 2018
%E A005253 Name clarified by _Jinyuan Wang_, Jan 20 2025