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 A003440 Number of binary vectors with restricted repetitions. (Formerly M2666) 6
 1, 1, 3, 7, 17, 42, 104, 259, 648, 1627, 4098, 10350, 26202, 66471, 168939, 430071, 1096451, 2799072, 7154189, 18305485, 46885179, 120195301, 308393558, 791882862, 2034836222, 5232250537, 13462265079, 34657740889, 89272680921, 230069128392 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The sum of squared terms in row n of A104402 = 2*a(n) for n>0. - Paul D. Hanna, Mar 06 2005 From Jean-Pierre Levrel, Nov 26 2014: (Start) The title "Binary Sequences with Restricted Repetitions," given the A003440 series, does not specify the type of restrictions used. After reading the article by K. A. Post, "Binary Sequences with Restricted Repetitions," it appears that the A003440 series corresponds to the following cases: - Number of repetitions limited to two, - Each sequence must begin with a zero. It is important to consider these two hypotheses to interpret the series. I also think that the second constraint is not useful and could usefully be deleted. In this case, the series should be doubled from the second term and would become 1, 2, 6, 14, 34, 84, ..., i.e., A177790. (End) REFERENCES 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 K. A. Post, Binary Sequences with Restricted Repetitions, Report 74-WSK-02, Math. Dept., Tech. Univ. Eindhoven, May. 1974. FORMULA G.f.: {(1-x)^2 * sqrt[(1+x+x^2)/(1-3x+x^2)] + x^2 - 1}/(2x^2) (conjectured). - Ralf Stephan, Mar 28 2004 a(n) = Sum_{k=0..n} (C(k, n-k) + C(k+1, n-k-1))^2/2 for n>0, with a(0)=1. - Paul D. Hanna, Mar 06 2005 Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(n-2)*a(n-2) +(-n+1)*a(n-3) +3*(n-4)*a(n-4) +(-n+5)*a(n-5)=0. - R. J. Mathar, Jun 07 2013 Recurrence: (n-2)*(n-1)*(n+2)*a(n) = 2*(n-2)*n*(n+1)*a(n-1) + (n-1)*(n^2 - 2*n - 4)*a(n-2) + 2*(n-3)*(n-2)*n*a(n-3) - (n-4)*(n-1)*n*a(n-4). - Vaclav Kotesovec, Feb 12 2014 a(n) ~ sqrt(6+14/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, Feb 12 2014 MATHEMATICA Flatten[{1, Table[Sum[(Binomial[k, n-k]+Binomial[k+1, n-k-1])^2/2, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 12 2014 *) a[r_, s_] /; r<0 || s<0 = 0; a[r_ /; 0 <= r <= 2, 0] = 1; a[r_ /; r>2, 0] = 0; a[0, s_ /; s >= 1] = 0; a[r_, s_] := a[r, s] = a[r-2, s-2] + a[r-2, s-1] + a[r-1, s-2] + a[r-1, s-1]; a[n_] := a[n, n]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jan 19 2015, after given recurrence *) PROG (PARI) {a(n)=polcoeff(((1-x)^2*sqrt((1+x+x^2)/(1-3*x+x^2))+x^2-1)/(2*x^2)+x*O(x^n), n)} \\ Paul D. Hanna, Mar 06 2005 (PARI) {a(n)=if(n==0, 1, sum(k=0, n, (binomial(k, n-k)+binomial(k+1, n-k-1))^2)/2)} \\ Paul D. Hanna, Mar 06 2005 CROSSREFS Cf. A078678, A104402, A177790. Sequence in context: A175094 A086395 A020730 * A244455 A102071 A191627 Adjacent sequences:  A003437 A003438 A003439 * A003441 A003442 A003443 KEYWORD nonn AUTHOR EXTENSIONS Typo in second formula corrected by Vaclav Kotesovec, Feb 12 2014 More terms from Vincenzo Librandi, Feb 13 2014 STATUS approved

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)