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A158869 Number of ways of filling a 2 X 3 X 2n hole with 1 X 2 X 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 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

Table of n, a(n) for n=0..19.

Martin Griffiths, Filling cuboidal holes with bricks, Mathematical Spectrum (Applied Probability Trust) 42(2) (2010), 91-92.

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).

a(n) = A138395(n+1) - A138395(n). (End)

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,changed

AUTHOR

Martin Griffiths, Mar 28 2009

EXTENSIONS

Edited by Charles R Greathouse IV, Mar 08 2011

STATUS

approved

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Last modified October 22 17:39 EDT 2019. Contains 328319 sequences. (Running on oeis4.)