

A158869


Number of ways of filling a 2 by 3 by 2*n hole with 1 by 2 by 2 bricks.


1



1, 5, 27, 147, 801, 4365, 23787, 129627, 706401, 3849525, 20977947, 114319107, 622980801, 3394927485, 18500622507, 100818952587, 549411848001, 2994014230245, 16315849837467, 88913056334067
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OFFSET

0,2


COMMENTS

Note that it is not possible to fill a 2 by 3 by (2*n1) hole using 1 by 2 by 2 bricks.
a(n+1) of the Jacobsthal sequence A001045 gives the number of ways of filling a 2 by 2 by n hole with 1 by 2 by 2 bricks.
Will the pattern of rightmost digits (1,5,7,7) be continued? [From 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. [From 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. [From Milan Janjic, Aug 13 2010]


REFERENCES

M. Griffiths, Filling cuboidal holes with bricks, Mathematical Spectrum (Applied Probability Trust), 42:2 (2010), pp. 9192.


LINKS

Table of n, a(n) for n=0..19.
Index entries for linear recurrences with constant coefficients, signature (6,3).


FORMULA

a(0)=1, a(1)=5 and a(n)=6*a(n1)3*a(n2) for n>1.
a(n) can be expressed in terms of Gauss' hypergeometric function as a(n)=(3^n) * 2F1[ ((n + 1)/2),(n/2),1/2,2/3].
From Martin Griffiths, Apr 02 2009: (Start)
G.f.: A(x)=(1x)/(16x+3x^2).
a(n)=(1/6)*((3+Sqrt[6])^(n+1)+(3Sqrt[6])^(n+1)). (End)
G.f.: (1+x)/(16*x+3*x^2). a(n)=A138395(n+1)A138395(n). [From R. J. Mathar, Mar 29 2009]
G.f.: G(0)/(6*x) 1/(3*x), where G(k)= 1 + 1/(1  x*(2*k3)/(x*(2*k1)  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

(Sage)
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

Sequence in context: A015535 A026292 A100193 * A162557 A134425 A184702
Adjacent sequences: A158866 A158867 A158868 * A158870 A158871 A158872


KEYWORD

easy,nonn


AUTHOR

Martin Griffiths, Mar 28 2009


EXTENSIONS

Edited by Charles R Greathouse IV, Mar 08, 2011


STATUS

approved



