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A055099 G.f.: (1+x)/(1-3*x-2*x^2). 30
1, 4, 14, 50, 178, 634, 2258, 8042, 28642, 102010, 363314, 1293962, 4608514, 16413466, 58457426, 208199210, 741512482, 2640935866, 9405832562, 33499369418, 119309773378, 424928058970, 1513403723666, 5390067288938 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums of triangle A054458.

a(n) = term (1,1) in M^n, M = the 3x3 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

REFERENCES

S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).

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, N. Murru, Construction and composition of rooted trees via descent functions, Algebra, Volume 2013 (2013), Article ID 543913, 11 pages.

A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics, 2 (1998), 197-210.

Shanzhen Gao, Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.

Sergey Kitaev, 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, 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(A054458(n, m), m=0..n). a(n) = F32(n)+F32(n-1) with F32(n) = A007482(n), n >= 1, a(0)=1.

a(n) = A007482(n)+A007482(n-1) = 2*A007482(n)-A104934(n). - R. J. Mathar, Jul 23 2010

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

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

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 *)

PROG

(Haskell)

a055099 n = a007481 (2 * n + 1) - a007481 (2 * n)

-- Reinhard Zumkeller, Oct 25 2015

CROSSREFS

Cf. A054458, A007482, A002203.

Cf. A001333. - Gary W. Adamson, Aug 14 2010

Column k=2 of A255256.

Cf. A007481.

Sequence in context: A034459 A120747 A229314 * A153367 A211304 A047008

Adjacent sequences:  A055096 A055097 A055098 * A055100 A055101 A055102

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang, Apr 26 2000

EXTENSIONS

Edited by N. J. A. Sloane, Jun 08 2010

STATUS

approved

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Last modified March 29 03:14 EDT 2017. Contains 284250 sequences.