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A023610
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Convolution of Fibonacci numbers and (F(2), F(3), F(4), ...).
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14
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1, 3, 7, 15, 30, 58, 109, 201, 365, 655, 1164, 2052, 3593, 6255, 10835, 18687, 32106, 54974, 93845, 159765, 271321, 459743, 777432, 1312200, 2211025, 3719643, 6248479, 10482351, 17562870, 29391490, 49132669, 82048737, 136884293
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n-2)+1 = number of (3412,1243)-, (3412,2134)- and (3412,1324)-avoiding involutions in S_n, n>1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 06 2003
The number of terms in all ordered partitions of (n+1) using only ones and twos. For example, a(3)=15 because there are 15 terms in 1+1+1+1;2+1+1;1+2+1;1+1+2;2+2 - Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 07 2008
a(n)=number of n-matchings in the graph obtained by a zig-zag triangulation of a convex (2n+1)-gon. Example: a(2)=7 because in the triangulation of the convex pentagon ABCDEA with diagonals AD and AC we have 7 2-matchings: {AB,CD},{AB,DE},{BC,AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2004
Partial sums of A029907. First differences of A002940. - Peter Bala (pbala(AT)toucansurf.com), Oct 24 2007
Equals row sums of triangle A144153. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2010: (Start)
Equals the number of 1's in Fibonacci Maximal notation for subsets of
(1, 2, 3, 5, 8, 13,...) terms. For example, (Cf. A181630): 4, 5, and 6 = the 3 terms 101, 110, and 111 in Fibonacci Maximal. Total number of 1's for those
terms = 7 = a(2). (End)
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REFERENCES
| M. Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
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LINKS
| E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8
Index to sequences with linear recurrences with constant coefficients, signature (2,1,-2,-1).
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FORMULA
| O.g.f.: (x+1)/(1-x-x^2)^2. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 11 2001
a(n) = 1/5*((n+2)F(n+4)+(n-1)F(n+2)), with F(n)=A000045(n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 06 2003
a(n)=sum{k=0..n+1, (n-k+1)*C(n-k+1, k)} - Paul Barry (pbarry(AT)wit.ie), Nov 05 2005
Recurrence: a(n+2) = a(n+1) + a(n) + Fib(n+4), n >= 0. For n >= 2, a(n-2) = (-1)^n*((-2n+3)*Fib(-n) - (-n)*Fib(-n-1))/5 = (-1)^n*A010049(-n), the second-order Fibonacci numbers of negative index, where Fib(-n) = (-1)^(n+1)*Fib(n). - Peter Bala (pbala(AT)toucansurf.com), Oct 24 2007
a(n) = (n+1)F(n+2) - A001629(n+1) where F(n) is the Fibonacci sequence - Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 07 2008
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MATHEMATICA
| Table[Sum[Binomial[n - i, i]*(n - i), {i, 0, n}], {n, 1, 33}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 04 2009]
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CROSSREFS
| Column 1 of triangle A063967.
Cf. A002940, A010049, A029907.
Cf. A144153 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
Cf. A181630 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2010]
Sequence in context: A058695 A187100 A182726 * A062544 A120411 A069112
Adjacent sequences: A023607 A023608 A023609 * A023611 A023612 A023613
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net), Jan 29 2004
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