A003229 a(n) = a(n-1) + 2*a(n-3) with a(0)=a(1)=1, a(2)=3.

1, 1, 3, 5, 7, 13, 23, 37, 63, 109, 183, 309, 527, 893, 1511, 2565, 4351, 7373, 12503, 21205, 35951, 60957, 103367, 175269, 297183, 503917, 854455, 1448821, 2456655, 4165565, 7063207, 11976517, 20307647, 34434061, 58387095, 99002389

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

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

References

• D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links below for an earlier version.
• 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
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.
• D. E. Daykin, Letter to N. J. A. Sloane, Dec 1973.
• D. E. Daykin, Letter to N. J. A. Sloane, Mar 1974.
• D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 1.
• D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 2.
• D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 3.
• D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, page 4.
• D. E. Daykin and S. J. Tucker, Sequences from Folding Paper, Unpublished manuscript, 1975, cached copy, reverse side of page 4.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 417.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (1,0,2).

Crossrefs

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

Cf. A155761, A003230.

Programs

Haskell

a003229 n = a003229_list !! n
a003229_list = 1 : 1 : 3 : zipWith (+)
                           (map (* 2) a003229_list) (drop 2 a003229_list)
-- Reinhard Zumkeller, Jan 01 2014

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

Maple

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

Mathematica

LinearRecurrence[{1,0, 2},{1,1,3},40] (* Vincenzo Librandi, Jun 12 2012 *)

Formula

G.f.: (1+2*z^2)/(1-z-2*z^3). - Simon Plouffe in his 1992 dissertation

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

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

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