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A032908
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One of 4 3rd-order recurring sequences for which the first derived sequence and the Galois transformed sequence coincide.
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3
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2, 2, 3, 6, 14, 35, 90, 234, 611, 1598, 4182, 10947, 28658, 75026, 196419, 514230, 1346270, 3524579, 9227466, 24157818, 63245987, 165580142, 433494438, 1134903171, 2971215074, 7778742050, 20365011075, 53316291174
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(n) is also a sequence with the property that the difference between the sum and product of two consecutive terms is equal to the square of the difference between those terms, i.e. a(n)*a(n+1) - (a(n)+ a(n+1)) = (a(n) - a(n + 1))^2. The difference between those two terms, a(n + 1) - a(n) = F(2n -2), the (2n - 2)th Fibonacci number. [From John Baker (john(AT)naturalmaths.com.au), May 18 2010]
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REFERENCES
| L. E. Dickson, History of the Theory of Numbers, Dover, New York, 1971
M. Elia, A Note on derived linear recurring sequences, pp. 83-92 of Proceedings Seventh Int. Conference on Fibonacci Numbers and their Applications (Austria, 1996), Applications of Fibonacci Numbers, Volume 7.
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LINKS
| Guo-Niu Han, Enumeration of Standard Puzzles
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 919
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FORMULA
| a(n) = 4*a(n-1) - 4*a(n-2)+a(n-3); g.f.: (2-6x+3x^2)/(1-4x+4x^2-x^3)
a(n) = Fibonacci(2*n-1)+1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 19 2003
a(n) = 3a(n - 1) - a(n - 2) - 1 [From Naoki Sato (nsato7(AT)yahoo.ca), Jan 21 2010]
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CROSSREFS
| Sequence in context: A095902 A103687 A166678 * A192366 A060631 A096100
Adjacent sequences: A032905 A032906 A032907 * A032909 A032910 A032911
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KEYWORD
| eigen,nonn
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AUTHOR
| Michele Elia (elia(AT)polito.it)
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EXTENSIONS
| More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 10 2003
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