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Search: id:A001871
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| A001871 |
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Expansion of 1/(1-3x+x^2 )^2.
(Formerly M4166 N1733)
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+0
5
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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; listen)
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OFFSET
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0,2
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COMMENT
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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
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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
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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.
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FORMULA
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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 kth Fibonacci number. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008
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MAPLE
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A001871:=1/(z**2-3*z+1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]
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PROGRAM
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(PARI) a(n)=((4*n+2)*fibonacci(2*n)+(7*n+5)*fibonacci(2*n+1))/5
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CROSSREFS
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Partial sums of A001870 (one half of odd indexed A001629(n), n >= 2, Fibonacci convolution).
Cf. A001629.
Sequence in context: A099625 A056279 A055337 this_sequence A000392 A099948 A092491
Adjacent sequences: A001868 A001869 A001870 this_sequence A001872 A001873 A001874
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KEYWORD
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nonn
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AUTHOR
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njas
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EXTENSIONS
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Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 07 2000
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