OFFSET
0,2
COMMENTS
Note that it is not possible to fill a 2 X 3 X (2*n-1) hole using 1 X 2 X 2 bricks.
a(n+1) of the Jacobsthal sequence A001045 gives the number of ways of filling a 2 X 2 X n hole with 1 X 2 X 2 bricks.
Will the pattern of rightmost digits (1,5,7,7) be continued? - Bill McEachen, May 20 2009
The answer to the question in a previous comment is: the linear recurrence proves that the pattern 1, 5, 7, 7 of the least significant digits will continue. - R. J. Mathar, Jun 20 2010
a(n) is the number of compositions of n when there are 5 types of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1358
Martin Griffiths, Filling cuboidal holes with bricks, Mathematical Spectrum (Applied Probability Trust) 42(2) (2010), 91-92.
Ryota Inagaki and Dimana Pramatarova, On Weighted and Bounded Multidimensional Catalan Numbers, arXiv:2510.14956 [math.CO], 2025. See p. 9.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 12.
Index entries for linear recurrences with constant coefficients, signature (6,-3).
FORMULA
a(0)=1, a(1)=5 and a(n) = 6*a(n-1) - 3*a(n-2) for n > 1.
a(n) = (3^n) * 2F1[-((n + 1)/2), -(n/2); 1/2; 2/3], using Gauss' hypergeometric function.
From Martin Griffiths, Apr 02 2009: (Start)
G.f.: A(x) = (1-x)/(1-6x+3x^2).
a(n) = (1/6)*((3+sqrt(6))^(n+1) + (3-sqrt(6))^(n+1)). (End)
From R. J. Mathar, Mar 29 2009: (Start)
G.f.: -(-1+x)/(1-6*x+3*x^2).
G.f.: G(0)/(6*x) -1/(3*x), where G(k) = 1 + 1/(1 - x*(2*k-3)/(x*(2*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2013
MATHEMATICA
Simplify[Table[ 1/6 * ((3 + Sqrt[6])^(n + 1) + (3 - Sqrt[6])^(n + 1)), {n, 0, 19}]]
Table[3^n * Hypergeometric2F1[ -((n + 1)/2), -(n/2), 1/2, 2/3], {n, 0, 19}]
LinearRecurrence[{6, -3}, {1, 5}, 30] (* Harvey P. Dale, May 28 2015 *)
PROG
(SageMath)
def A158869(n): return 3^n*lucas_number2(n+1, 2, 1/3)/2
[A158869(n) for n in (0..19)] # Peter Luschny, May 06 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Martin Griffiths, Mar 28 2009
EXTENSIONS
Edited by Charles R Greathouse IV, Mar 08 2011
STATUS
approved