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A001871
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Expansion of 1/(1-3x+x^2 )^2.
(Formerly M4166 N1733)
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9
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1, 6, 25, 90, 300, 954, 2939, 8850, 26195, 76500, 221016, 632916, 1799125, 5082270, 14279725, 39935214, 111228804, 308681550, 853904015, 2355364650, 6480104231, 17786356776, 48715278000, 133167004200, 363372003625, 989900286774
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) = (2*(2*n+1)*F(2*(n+1))+3*(n+1)*F(2*n+1))/5 with F(n) = A000045 (Fibonacci numbers).
Convolution of A001906(n), n >= 1, (even indexed Fibonacci numbers) with itself.
a(n) = -a(-4-n) = ((4n+2)F(2n)+(7n+5)F(2n+1))/5 with F(n) = A000045 (Fibonacci numbers).
A001787 and this sequence arise in counting ordered trees of height at most k where only the right-most branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for A001787 and k = 4 for this sequence.
Gives the number of 3412-avoiding permutations containing exactly one subsequence of type 321. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008
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REFERENCES
| N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences
Index to sequences with linear recurrences with constant coefficients, signature (6,-11,6,-1).
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FORMULA
| a(n) = [2a(n-1)+(n+1)F(2n+4)]/3, where F(n) = A000045 (Fibonacci numbers). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 08 2002
G.f.: 1/(1-3x+x^2)^2.
a(n) = sum{k = 0..n, S(k, 3)S(n-k, 3)} S(n, x) = U(n, x/2) Chebyshev polynomials of 2nd kind, A049310 - Paul Barry (pbarry(AT)wit.ie), Nov 14 2003
a(n)= sum_{k=1}^{n+1} F(2k)F(2(n-k+2)) where F(k) is the k-th Fibonacci number. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). Vincenzo Librandi, Mar 14 2011
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MAPLE
| A001871:=1/(z**2-3*z+1)**2; [S. Plouffe in his 1992 dissertation.]
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PROG
| (PARI) a(n)=((4*n+2)*fibonacci(2*n)+(7*n+5)*fibonacci(2*n+1))/5
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CROSSREFS
| Partial sums of A001870 (one half of odd indexed A001629(n), n >= 2, Fibonacci convolution).
Cf. A001629.
Sequence in context: A143628 A056279 A055337 * A000392 A099948 A143815
Adjacent sequences: A001868 A001869 A001870 * A001872 A001873 A001874
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 07 2000
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