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A001911
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Fibonacci(n+3) - 2.
(Formerly M2546 N1007)
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51
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0, 1, 3, 6, 11, 19, 32, 53, 87, 142, 231, 375, 608, 985, 1595, 2582, 4179, 6763, 10944, 17709, 28655, 46366, 75023, 121391, 196416, 317809, 514227, 832038, 1346267, 2178307, 3524576, 5702885, 9227463, 14930350, 24157815, 39088167, 63245984
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This is the sequence A(0,1;1,1;2)of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [From Wolfdieter Lang, Oct 17 2010]
Contribution from Gary W. Adamson, Nov 02 2010: (Start)
For subsets of (1, 2, 3, 5, 8, 13,...) Fibonacci Maximal terms (Cf. A181631)
equals the number of leading 1's per subset. For example, (7-11) in Fibonacci
Maximal = (1010, 1011, 1101, 1110, 1111), numbers of leading 1's =
(1 + 1 + 2 + 3 + 4) = 11 = a(4) = row 4 of triangle A181631. (End)
As in the reference arXiv.org/abs/0910.4432 we use two types of Fibonacci trees:- Ta: Fibonacci analog of binomial trees; Tb: Binary Fibonacci trees. Let D(r(k)) be the sum over all distances of the form d(r,x), across all vertices x of the tree rooted at r of order k. Ignoring r, but overloading, let D(a(k)) and D(b(k)) be the distance sums for the Fibonacci trees Ta and Tb respectively of the order k. Using the sum-of-product form in the eqs.(18) and (21) in the said reference it follows that F(k+4)-2 = D(a(k+1))-D(b(k-1)). [Kailasam Viswanathan Iyer and P. Venkata Subba Reddy, Apr 30 2011.]
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REFERENCES
| J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233.
D. G. Rogers, An application of renewal sequences to the dimer problem, pp. 142-153 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
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).
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LINKS
| D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory
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.
W. Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 17 2010]
K. Viswanathan Iyer and K. R. Uday Kumar Reddy, Wiener index of binomial trees and Fibonacci trees (Corrigendum: Eq.(23) to be corrected as folows on the right-side: in the fourth term F(k)-1 should be replaced by F(k); a term F(k)*F(K+1)-1 is to be included; pointed out by Emeric Deutsch).
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1)
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FORMULA
| a(n) = a(n-1) + a(n-2) + 2, a(0)=0, a(1)=1.
G.f.: x*(1+x)/((1-x)*(1-x-x^2)).
Sum of consecutive pairs of A000071 (partial sums of Fibonacci numbers). - Paul Barry (pbarry(AT)wit.ie), Apr 17 2004
a(n) = A101220(2, 1, n) - Ross La Haye, Jan 28 2005
a(n) = A108617(n+1, 2) = A108617(n+1, n-1) for n>0; - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) in the 1x3 matrix [0,-1,1].[1,1,0; 1,0,0; 2,0,1]^n. - Alois P. Heinz, Jul 24 2008
a(0)=0, a(1)=1, a(2)=3, a(n)=2*a(n-1)-a(n-3) [From Harvey P. Dale, June 06 2011]
Eigensequence of an infinite lower triangular matrix with the natural numbers as the left border and (1, 0, 1, 0,...) in all other columns. - Gary W. Adamson, Jan 30 2012
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MAPLE
| a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+2 od: seq(a[n], n=0..50); (Miklos Kristof, Mar 09 2005)
A001911:=(1+z)/(z-1)/(z**2+z-1); [S. Plouffe in his 1992 dissertation with another offset.]
a := n -> (Matrix([[0, -1, 1]]) . Matrix([[1, 1, 0], [1, 0, 0], [2, 0, 1]])^n)[1, 1]; seq (a(n), n=0..50); - Alois P. Heinz, Jul 24 2008
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MATHEMATICA
| Table[Fibonacci[n+3]-2, {n, 0, 5!}] [From Vladimir Joseph Stephan Orlovsky, Nov 19 2010]
LinearRecurrence[{2, 0, -1}, {0, 1, 3}, 40] (* From Harvey P. Dale, June 06 2011 *)
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PROG
| (MAGMA) [(Fibonacci(n+3))-2: n in [0..85]]; // Vincenzo Librandi, Apr 23 2011
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CROSSREFS
| a(n) = A000045(n+3)-2.
Partial sums of F(n+1)=A000045(n+1).
Right-hand column 3 of triangle A011794.
Cf. A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by N. J. A. Sloane, Jun 25 2010 in response to a comment from Aviezri S. Fraenkel]
Cf. A181631 [From Gary W. Adamson, Nov 02 2010]
see also A165910.
Sequence in context: A144115 A183088 A116557 * A020957 A179006 A191696
Adjacent sequences: A001908 A001909 A001910 * A001912 A001913 A001914
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms and better description from Michael Somos
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