OFFSET
0,4
COMMENTS
Trisection of the Padovan sequence: a(n) = A000931(3n). - Paul Barry, Jul 06 2004
a(n+1) gives diagonal sums of Riordan array (1/(1-x),x/(1-x)^3). - Paul Barry, Oct 11 2005
a(n+2) is the sum, over all Boolean n-strings, of the product of the lengths of the runs of 1. For example, the Boolean 7-string (0,1,1,0,1,1,1) has two runs of 1s. Their lengths, 2 and 3, contribute a product of 6 to a(9). The 8 Boolean 3-strings contribute to a(5) as follows: 000 (empty product), 001, 010, 100, 101 all contribute 1, 011 and 110 contribute 2, 111 contributes 3. - David Callan, Nov 29 2007
[a(n), a(n+1), a(n+2)], n > 0, = [0,1,0; 0,0,1; 1,-2,3]^n * [1,1,1]. - Gary W. Adamson, Mar 27 2008
Without the initial 1 and 1: 1, 2, 5, 12, 28, this is also the transform of 1 by the T_{1,0} transformation; see Choulet link. - Richard Choulet, Apr 11 2009
Without the first 1: transform of 1 by T_{0,0} transformation (see Choulet link). - Richard Choulet, Apr 11 2009
Starting (1, 2, 5, 12, ...) = INVERT transform of (1, 1, 2, 3, 4, 5, ...) and row sums of triangle A159974. - Gary W. Adamson, Apr 28 2009
a(n+1) is also the number of 321-avoiding separable permutations. (A permutation is separable if it avoids both 2413 and 3142.) - Vince Vatter, Sep 21 2009
a(n+1) is an eigensequence of the sequence array for (1,1,2,3,4,5,...). - Paul Barry, Nov 03 2010
Equals the INVERTi transform of A055588: (1, 2, 4, 9, 22, 56, ...) - Gary W. Adamson, Apr 01 2011
The Ca3 sums, see A180662, of triangle A194005 equal the terms of this sequence without a(0) and a(1). - Johannes W. Meijer, Aug 16 2011
Without the initial 1, a(n) = row sums of A182097(n)*A007318(n,k); i.e., a Triangular array T(n,k) multiplying the binomial (Pascal's) triangle by the Padovan sequence where a(0) = 1, a(1) = 0 and a(2) = 1. - Bob Selcoe, Jun 28 2013
a(n+1) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 0, 1, 1; 1, 0, 1] or [1, 1, 0; 1, 1, 1; 1, 0, 1] or [1, 1, 1; 1, 1, 0; 0, 1, 1] or [1, 0, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 0, 1; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 1, 0; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
Number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k) and e(i) <= e(k). [Martinez and Savage, 2.8] - Eric M. Schmidt, Jul 17 2017
a(n+1) is the number of words of length n over the alphabet {0,1,2} that do not contain the substrings 01 or 12 and do not start with a 2 and do not end with a 0. - Yiseth K. RodrÃguez C., Sep 11 2020
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Miklos Bona and Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See H(z), Ex. 4.1.
Richard Choulet, Curtz like Transformation
Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From N. J. A. Sloane, Feb 01 2013
Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 904
H. Magnusson and H. Ulfarsson, Algorithms for discovering and proving theorems about permutation patterns, arXiv preprint arXiv:1211.7110 [math.CO], 2012.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016
Vincent Vatter, Finding regular insertion encodings for permutation classes, arXiv:0911.2683 [math.CO], 2009.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).
FORMULA
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1, 3*k). - Paul Barry, Jul 06 2004
G.f.: (1 - 2*x)/(1 - 3*x + 2*x^2 - x^3). - Paul Barry, Jul 06 2005
G.f.: 1 + x / (1 - x / (1 - x / (1 - x / (1 + x / (1 - x))))). - Michael Somos, Mar 31 2012
a(-1 - n) = A185963(n). - Michael Somos, Mar 31 2012
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 28*x^6 + 65*x^7 + 151*x^8 + ...
MAPLE
A034943 := proc(n): add(binomial(n+k-1, 3*k), k=0..floor(n/2)) end: seq(A034943(n), n=0..28); # Johannes W. Meijer, Aug 16 2011
MATHEMATICA
LinearRecurrence[{3, -2, 1}, {1, 1, 1}, 30] (* Harvey P. Dale, Aug 11 2017 *)
PROG
(Magma) [n le 3 select 1 else 3*Self(n-1)-2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012
(PARI) {a(n) = if( n<1, n = 0-n; polcoeff( (1 - x + x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n), n = n-1; polcoeff( (1 - x + x^2) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Mar 31 2012 */
(SageMath)
@CachedFunction
def a(n): # a = A034943
if (n<3): return 1
else: return 3*a(n-1) - 2*a(n-2) + a(n-3)
[a(n) for n in range(51)] # G. C. Greubel, Apr 22 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Charles R Greathouse IV, Apr 20 2010
STATUS
approved