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A022026
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Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(2,15).
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12
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2, 15, 112, 836, 6240, 46576, 347648, 2594880, 19368448, 144568064, 1079070720, 8054293504, 60118065152, 448727347200, 3349346516992, 24999862747136, 186601515909120, 1392812676284416, 10396095346638848, 77597512067973120, 579195715157229568
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OFFSET
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0,1
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COMMENTS
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a(n) is also the number of forests in the 2 X (n+1) grid.
a(0)=2, because there are 2 forests in the 2 X 1 grid: 1.2 and 1-2.
a(1)=15, because there are 15 forests in the 2 X 2 grid:
1.2 1-2 1.2 1.2 1.2 1-2 1-2 1-2 1.2 1.2 1.2 1.2 1-2 1-2 1-2
. . . . . | . . | . . | . . | . . | | | | . | | | . | | . |
4.3 4.3 4.3 4-3 4.3 4.3 4-3 4.3 4-3 4.3 4-3 4-3 4-3 4.3 4-3
a(n) = 8a(n-1) - 4a(n-2) for n>=2, because each of the a(n-1) forests can be extended by 8 patterns:
.o -o .o -o .o -o .o -o
.. .. .. .. .| .| .| .|
.o .o -o -o .o .o -o -o
where 4a(n-2) of these are not forests, namely the extensions of a(n-2) forests by 4 patterns:
.o-o -o-o .o-o -o-o
.| | .| | .| | .| |
.o-o .o-o -o-o -o-o (End)
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LINKS
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FORMULA
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G.f.: (2-x)/(1-8x+4x^2). - David Boyd and Ralf Stephan, Apr 15 2004
a(n) = sum of the entries in the 2 X 2 matrix A^n where A is the 2 X 2 matrix [4, 4; 3, 4].
a(n) = (1 + 7*sqrt(3)/12)*(4 + 2*sqrt(3))^n + (1 - 7*sqrt(3)/12)*(4 - 2*sqrt(3))^n. See Desjarlais and Molina. (End)
a(n+1) = ceiling(a(n)^2/a(n-1))-1 for all n > 0. - M. F. Hasler, Feb 10 2016
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MAPLE
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a:= n-> (Matrix([[15, 2]]). Matrix([[8, 1], [-4, 0]])^n)[1, 2]:
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MATHEMATICA
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CoefficientList[Series[(2-x)/(1-8*x+4*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, May 03 2014 *)
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PROG
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(PARI) a(n)=([15, 2]*[8, 1; -4, 0]^n)[2] \\ M. F. Hasler, Feb 10 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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