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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Note that it is not possible to fill a 2 by 3 by (2*n-1) 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. 91-92.

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(n-1)-3*a(n-2) 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)=(1-x)/(1-6x+3x^2).

a(n)=(1/6)*((3+Sqrt[6])^(n+1)+(3-Sqrt[6])^(n+1)). (End)

G.f.: -(-1+x)/(1-6*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*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

(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 A305573

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

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Last modified December 12 00:07 EST 2018. Contains 318052 sequences. (Running on oeis4.)