%I M2419 #107 Aug 01 2024 03:03:24

%S 1,1,3,5,7,13,23,37,63,109,183,309,527,893,1511,2565,4351,7373,12503,

%T 21205,35951,60957,103367,175269,297183,503917,854455,1448821,2456655,

%U 4165565,7063207,11976517,20307647,34434061,58387095,99002389

%N a(n) = a(n-1) + 2*a(n-3) with a(0)=a(1)=1, a(2)=3.

%C Equals eigensequence of an infinite lower triangular matrix with 1's in the main diagonal, 0's in the subdiagonal and 2's in the subsubdiagonal (the triangle in the lower section of A155761). - _Gary W. Adamson_, Jan 28 2009

%C The operation in the comment of Jan 28 2009 is equivalent to the INVERT transform of (1, 0, 2, 0, 0, 0, ...). - _Gary W. Adamson_, Jan 21 2017

%C For n>=1, a(n) equals the number of ternary words of length n-1 having at least 2 zeros between every two successive nonzero letters. - _Milan Janjic_, Mar 09 2015

%D D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links below for an earlier version.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A003229/b003229.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.

%H D. E. Daykin, <a href="/A003229/a003229_1.pdf">Letter to N. J. A. Sloane, Dec 1973</a>.

%H D. E. Daykin, <a href="/A003229/a003229.pdf">Letter to N. J. A. Sloane, Mar 1974</a>.

%H D. E. Daykin and S. J. Tucker, <a href="/A003229/a003229.jpg">Sequences from Folding Paper</a>, Unpublished manuscript, 1975, cached copy, page 1.

%H D. E. Daykin and S. J. Tucker, <a href="/A003229/a003229_1.jpg">Sequences from Folding Paper</a>, Unpublished manuscript, 1975, cached copy, page 2.

%H D. E. Daykin and S. J. Tucker, <a href="/A003229/a003229_2.jpg">Sequences from Folding Paper</a>, Unpublished manuscript, 1975, cached copy, page 3.

%H D. E. Daykin and S. J. Tucker, <a href="/A003229/a003229_3.jpg">Sequences from Folding Paper</a>, Unpublished manuscript, 1975, cached copy, page 4.

%H D. E. Daykin and S. J. Tucker, <a href="/A003229/a003229_4.jpg">Sequences from Folding Paper</a>, Unpublished manuscript, 1975, cached copy, reverse side of page 4.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=417">Encyclopedia of Combinatorial Structures 417</a>.

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2).

%F G.f.: (1+2*z^2)/(1-z-2*z^3). - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = A077949(n+1) = |A077974(n+1)|.

%F a(n) = u(n+1) - 3*u(n) + 2*u(n-1) where u(i) = A003230(i) [Daykin and Tucker]. - _N. J. A. Sloane_, Jul 08 2014

%F a(n) = hypergeom([-n/3,-(1+n)/3,(1-n)/3],[-n/2,-(1+n)/2],-27/2) for n>=3. - _Peter Luschny_, Mar 09 2015

%p seq(add(binomial(n-2*k,k)*2^k,k=0..floor(n/3)),n=1..38); # _Zerinvary Lajos_, Apr 03 2007

%p with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Sequence(Z, card >=3)}, unlabeled]: seq(count(SeqSeqSeqL, size=n+4), n=0..35); # _Zerinvary Lajos_, Apr 04 2009

%p a := n -> `if`(n<3,[1,1,3][n+1], hypergeom([-n/3, -(1+n)/3, (1-n)/3], [-n/2, -(1+n)/2], -27/2)); seq(simplify(a(n)),n=0..35); # _Peter Luschny_, Mar 09 2015

%t LinearRecurrence[{1,0, 2},{1,1,3},40] (* _Vincenzo Librandi_, Jun 12 2012 *)

%o (Magma) I:=[1, 1, 3]; [n le 3 select I[n] else Self(n-1)+2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jun 12 2012

%o (Haskell)

%o a003229 n = a003229_list !! n

%o a003229_list = 1 : 1 : 3 : zipWith (+)

%o (map (* 2) a003229_list) (drop 2 a003229_list)

%o -- _Reinhard Zumkeller_, Jan 01 2014

%Y Cf. A077949, A077974. First differences of A003479. Partial sums of A052537. Equals |A077906(n)|+|A077906(n+1)|.

%Y Cf. A155761, A003230.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_