

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
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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 JeanPierre 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 74WSK02, Math. Dept., Tech. Univ. Eindhoven, May. 1974.


FORMULA

G.f.: {(1x)^2 * sqrt[(1+x+x^2)/(13x+x^2)] + x^2  1}/(2x^2) (conjectured).  Ralf Stephan, Mar 28 2004
a(n) = Sum_{k=0..n} (C(k, nk) + C(k+1, nk1))^2/2 for n>0, with a(0)=1.  Paul D. Hanna, Mar 06 2005
Conjecture: (n+2)*a(n) +3*(n1)*a(n1) +(n2)*a(n2) +(n+1)*a(n3) +3*(n4)*a(n4) +(n+5)*a(n5)=0.  R. J. Mathar, Jun 07 2013
Recurrence: (n2)*(n1)*(n+2)*a(n) = 2*(n2)*n*(n+1)*a(n1) + (n1)*(n^2  2*n  4)*a(n2) + 2*(n3)*(n2)*n*a(n3)  (n4)*(n1)*n*a(n4).  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
Equivalently, a(n) ~ phi^(2*n + 2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio.  Vaclav Kotesovec, Dec 08 2021


MATHEMATICA

Flatten[{1, Table[Sum[(Binomial[k, nk]+Binomial[k+1, nk1])^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[r2, s2] + a[r2, s1] + a[r1, s2] + a[r1, s1]; a[n_] := a[n, n]; Table[a[n], {n, 0, 29}] (* JeanFrançois Alcover, Jan 19 2015, after given recurrence *)


PROG

(PARI) {a(n)=polcoeff(((1x)^2*sqrt((1+x+x^2)/(13*x+x^2))+x^21)/(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, nk)+binomial(k+1, nk1))^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

N. J. A. Sloane


EXTENSIONS

Typo in second formula corrected by Vaclav Kotesovec, Feb 12 2014
More terms from Vincenzo Librandi, Feb 13 2014


STATUS

approved



