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A158869
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Number of ways of filling a 2 by 3 by 2*n hole with 1 by 2 by 2 bricks.
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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; internal format)
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OFFSET
| 0,2
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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 continue? [From Bill R McEachen (bmceache(AT)centralsan.org), 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 (mathar(AT)strw.leidenuniv.nl), 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 R. Janjic (agnus(AT)blic.net), Aug 13 2010]
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REFERENCES
| M. Griffiths, Filling cuboidal holes with bricks, Mathematical Spectrum (Applied Probability Trust), 42:2 (2010), pp. 91-92.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (6,-3).
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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].
Contribution from Martin Griffiths (griffm(AT)essex.ac.uk), 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 (mathar(AT)strw.leidenuniv.nl), Mar 29 2009]
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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}]
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CROSSREFS
| Sequence in context: A015535 A026292 A100193 * A162557 A134425 A184702
Adjacent sequences: A158866 A158867 A158868 * A158870 A158871 A158872
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KEYWORD
| easy,nonn
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AUTHOR
| Martin Griffiths (griffm(AT)essex.ac.uk), Mar 28 2009
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EXTENSIONS
| Edited by Charles R Greathouse IV, Mar 08, 2011
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