OFFSET
0,2
COMMENTS
Row sums of triangle A054458.
a(n) = term (1,1) in M^n, M = the 3 X 3 matrix [1,1,1; 1,1,1; 2,2,1]. - Gary W. Adamson, Mar 12 2009
Equals the INVERT transform of A001333: (1, 3, 7, 17, 41, 99, ...). - Gary W. Adamson, Aug 14 2010
a(n) is the number of one sided n-step walks taking steps from {(0,1), (-1,0), (1,0), (1,1)}. - Shanzhen Gao, May 13 2011
Number of quaternary words of length n on {0,1,2,3} containing no subwords 03 or 30. - Philippe Deléham, Apr 27 2012
Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 2, 272, 12, 18, 24, ... - R. J. Mathar, Aug 10 2012
a(n) = A007481(2*n+1) - A007481(2*n) = A007481(2*(n+1)) - A007481(2*n+1). - Reinhard Zumkeller, Oct 25 2015
Number of length-n words on a,b,c,d avoiding aa and ab. For n >= 1, the number of such words ending with a or the number of those ending with b is A007482(n-1), and the number of those ending with c or the number of those ending with d is a(n-1). - Jianing Song, Jun 01 2022
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.4.6).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. Abrate, S. Barbero, U. Cerruti, and N. Murru, Construction and composition of rooted trees via descent functions, Algebra, Volume 2013 (2013), Article ID 543913, 11 pages.
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 17
A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Sergey Kitaev and Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (3,2).
FORMULA
a(n) = a*c^n - b*d^n, a := (5 + sqrt(17))/(2*sqrt(17)), b := (5 - sqrt(17))/(2*sqrt(17)), c := (3 + sqrt(17))/2, d := (3 - sqrt(17))/2.
a(n) = Sum_{m=0..n} A054458(n, m).
a(n) = F32(n) + F32(n-1) with F32(n) = A007482(n), n >= 1, a(0) = 1.
a(n) = 3*a(n-1) + 2*a(n-2) with a(0) = 1, a(1) = 4. - Vincenzo Librandi, Dec 08 2010
a(n) = (Sum_{k = 0..n} A202396(n,k)*3^k)/2^n. - Philippe Deléham, Feb 05 2012
a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - G. C. Greubel, Jun 27 2021
a(n) = 2*a(n-1) + 2*A007482(n-1), n >= 1. - Jianing Song, Jun 01 2022
E.g.f.: exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, May 24 2024
EXAMPLE
a(3) = 50 because among the 4^3 = 64 quaternary words of length 3 only 14 namely 003, 030, 031, 032, 033, 103, 130, 203, 230, 300, 301, 302, 303, 330 contain the subwords 03 or 30. - Philippe Deléham, Apr 27 2012
MAPLE
a := proc(n) option remember; `if`(n < 2, [1, 4][n+1], (3*a(n-1) + 2*a(n-2))) end:
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2019
MATHEMATICA
max = 24; cv = ContinuedFraction[ Sqrt[2], max] // Convergents // Numerator; Series[ 1/(1 - cv.x^Range[max]), {x, 0, max}] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Jun 21 2013, after Gary W. Adamson *)
LinearRecurrence[{3, 2}, {1, 4}, 24] (* Jean-François Alcover, Sep 23 2017 *)
PROG
(Haskell)
a055099 n = a007481 (2 * n + 1) - a007481 (2 * n)
-- Reinhard Zumkeller, Oct 25 2015
(Magma) I:=[1, 4]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jun 27 2021
(Sage) [(i*sqrt(2))^(n-1)*( i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # G. C. Greubel, Jun 27 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Apr 26 2000
EXTENSIONS
Edited by N. J. A. Sloane, Jun 08 2010
STATUS
approved