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A000071 Fibonacci numbers - 1.
(Formerly M1056 N0397)
188
0, 0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308, 3524577, 5702886, 9227464, 14930351, 24157816, 39088168 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n) is the number of allowable transition rules for passing from one change to the next (on n-1 bells) in the English art of bell-ringing. This is also the number of involutions in the symmetric group S_{n-1} which can be represented as a product of transpositions of consecutive numbers from {1,2,...,n-1}. Thus for n=6 we have a(6) = from (12), (12)(34), (12)(45), (23), (23)(45), (34), (45), for instance. See my 1983 Math. Proc. Camb. Phil. Soc. paper. - Arthur T. White, letter to N. J. A. Sloane, Dec 18 1986

Number of permutations p of {1,2,...,n-1} such that max|p(i)-i|=1. Example: a(4)=2 since only the permutations 132 and 213 of {1,2,3} satisfy the given condition. - Emeric Deutsch, Jun 04 2003

Number of 001-avoiding binary words of length n-3.

Also, sum of first n Fibonacci numbers. - Giorgi Dalakishvili (mcnamara_gio(AT)yahoo.com), Apr 02 2005

a(n)=number of partitions of {1,...,n-1} into two blocks in which only 1- or 2-strings of consecutive integers can appear in a block and there is at least one 2-string. E.g. a(6) = 7 because the enumerated partitions of {1,2,3,4,5} are 124/35,134/25, 14/235,13/245,1245/3,145/23,125/34. - Augustine O. Munagi, Apr 11 2005

Numbers for which only one Fibonacci bit-representation is possible and for which the maximal and minimal Fibonacci bit-representations (A104326 and A014417) are equal. For example, a(12) = 10101 because 8+3+1 = 12. - Casey Mongoven, Mar 19 2006

Beginning with a(2), the 'Recamán transform' (see A005132) of the Fibonacci numbers (A000045). - Nick Hobson (nickh(AT)qbyte.org), Mar 01 2007

Starting with nonzero terms = row sums of triangle A158950. - Gary W. Adamson, Mar 31 2009

a(n+2) = minimum number of elements in an AVL tree of height n. - Lennert Buytenhek (buytenh(AT)wantstofly.org), May 31 2010

a(n) is the number of branch nodes in the Fibonacci tree of order n-1. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node (see the Knuth reference, p. 417). - Emeric Deutsch, Jun 14 2010

a(n+3) is the number of distinct three-strand positive braids of length n (cf. Burckel). - Maxime Bourrigan, Apr 04 2011

A000119(a(n)) = 1. - Reinhard Zumkeller, Dec 28 2012

a(n+1) is the number of compositions of n with maximal part 2. - Joerg Arndt, May 21 2013

a(n) = A228074(n-1,2) for n > 2. - Reinhard Zumkeller, Aug 15 2013

a(0) = -1. For a(5) = 4 we have 2143, 1324, 2134 and 1243. - Jon Perry, Sep 14 2013

a(n+2) is the number of leafs of great-grandparent DAG (directed acyclic graph) of height n. A great-grandparent DAG of height n is a single node for n=1; for n>1 each leaf of ggpDAG(n-1) has two child nodes where pairs of adjacent new nodes are merged into single node iff they have disjoint grandparents and same great-grandparent. Consequence: a(n) = 2*a(n-1) - a(n-3). - Hermann Stamm-Wilbrandt, Jul 06 2014

A083368(a(n+3)) = n. - Reinhard Zumkeller, Aug 10 2014

REFERENCES

Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.

A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014, http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 1.

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 64.

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. [From Emeric Deutsch, Jun 14 2010]

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 155.

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).

J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.

LINKS

Christian G. Bower, Table of n, a(n) for n = 1..500

J.-L. Baril, J.-M. Pallo, Motzkin subposet and Motzkin geodesics in Tamari lattices/a>, 2013.

S. Burckel, Syntactical methods for braids of three strands, J. Symbolic Comput. 31 (2001), no. 5, 557-564.

A. Burstein and T. Mansour, Counting occurrences of some subword patterns.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Fan Chung, Ron Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194

Dairyko, Michael; Tyner, Samantha; Pudwell, Lara; Wynn, Casey. Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From N. J. A. Sloane, Feb 01 2013

Emeric Deutsch, Problem Q915, Math. Magazine, vol. 74, No. 5, 2001, p. 404.

Fumio Hazama, Spectra of graphs attached to the space of melodies, Discrete Math., 311 (2011), 2368-2383. See Table 2.1.

Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [From Emeric Deutsch, Jun 14 2010]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 384

R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'Ecole Normale Supérieure, Paris, 79 (1962), 199-241.

D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.

Rui Liu and Feng-Zhen Zhao, On the Sums of Reciprocal Hyperfibonacci Numbers and Hyperlucas Numbers, Journal of Integer Sequences, Vol. 15 (2012), #12.4.5. - From N. J. A. Sloane, Oct 05 2012

A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005), 451-463.

Sam Northshield, Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,..., Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

L. Pudwell, Pattern avoidance in trees, (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013

Stacey Wagner, Enumerating Alternating Permutations with One Alternating Descent, DePaul Discoveries: Vol. 2: Iss. 1, Article 2.

Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.

P. Xu, Growth of positive braids semigroups, Journal of Pure and Applied Algebra, 1992.

Index entries for sequences related to linear recurrences with constant coefficients, signature (2,0,-1).

FORMULA

a(n) = A000045(n) - 1.

a(0) = -1, a(1) = 0; thereafter a(n) = a(n-1)+a(n-2)+1.

a(n) = 2*a(n-1)-a(n-3). - R. H. Hardin, Apr 02 2011

Partial sums of Fibonacci numbers. G.f.: x^3/((1-x-x^2)*(1-x)). - Wolfdieter Lang

a(n) = -1+(A*B^n+C*D^n)/10, with A, C=5+-3*sqrt(5), B, D=(1+-sqrt(5))/2. - Ralf Stephan, Mar 02 2003

a(1)=0, a(2)=0, a(3)=1, then a(n)=ceiling(phi*a(n-1)) where phi is the golden ratio (1+sqrt(5))/2. - Benoit Cloitre, May 06 2003

Conjecture: for all c such that 2*(2-Phi) <= c < (2+Phi)*(2-Phi) we have a(n) = floor(Phi*a(n-1)+c) for n > 3. - Gerald McGarvey, Jul 22 2004

a(n) = sum{k=0..floor((n-2)/2), binomial(n-k-2, k+1)}. - Paul Barry, Sep 23 2004

a(n+3) = sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*2^(n-3k)}. - Paul Barry, Oct 20 2004

a(n+1) = Sum(binomial(n-r, r)), r=1, 2, ... which is the case t=2 and k=2 in the general case of t-strings and k blocks: a(n+1, k, t)=Sum(binomial(n-r*(t-1), r)*S2(n-r*(t-1)-1, k-1)), r=1, 2, ... - Augustine O. Munagi, Apr 11 2005

a(n) = Sum[k*Fibonacci(n-k-3),{k,0,n-2}]. - Ross La Haye, May 31 2006

a(n) = term (3,2) in the 3x3 matrix [1,1,0; 1,0,0; 1,0,1]^(n-1). - Alois P. Heinz, Jul 24 2008

For n>=4, a(n)=ceil(phi*a(n-1)), where phi=Golden ratio. - Vladimir Shevelev, Jul 04 2010

Closed-form without two leading zeros G.f.: 1/(1-2*x-x^3); ((5+2*sqrt(5))*((1+sqrt(5))/2)^n + (5-2*sqrt(5))*((1-sqrt(5))/2)^n - 5)/5; closed-form with two leading zeros G.f.: x^2/(1-2*x-x^3); ((5+sqrt(5))*((1+sqrt(5))/2)^n + (5-sqrt(5))*((1-sqrt(5))/2)^n - 10)/10. - Tim Monahan, Jul 10 2011

G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - x^2)/( x*(4*k+4 - x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013

MAPLE

a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+1 od: seq(a[n], n=0..50); # Kristof

A000071 := proc(n) combinat[fibonacci](n)-1 ; end proc; # R. J. Mathar, Apr 07 2011

A000071:=1/(z-1)/(z^2+z-1); # Simon Plouffe in his 1992 dissertation, dropping initial zeros

a:= n-> (Matrix ([[1, 1, 0], [1, 0, 0], [1, 0, 1]])^(n-1))[3, 2]; seq(a(n), n=1..50); # Alois P. Heinz, Jul 24 2008

MATHEMATICA

Table[f=Fibonacci[k]; f-1, {k, 1, 40, 1}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)

Fibonacci[Range[40]]-1 (* or *) LinearRecurrence[{2, 0, -1}, {0, 0, 1}, 40] (* Harvey P. Dale, Aug 23 2013 *)

PROG

(PARI) {a(n) = if( n<1, 0, fibonacci(n)-1)};

(MAGMA) [Fibonacci(n)-1: n in [1..150]] // Vincenzo Librandi, Apr 04 2011

(Haskell)

a000071 n = a000071_list !! n

a000071_list = map (subtract 1) $ tail a000045_list

-- Reinhard Zumkeller, May 23 2013

CROSSREFS

Cf. A000045, A054761, A119282, A001654, A005968, A005969, A098531, A098532, A098533, A128697, A001611, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616, A158950, A105488, A105489.

Antidiagonal sums of array A004070.

Right-hand column 2 of triangle A011794.

a(n) = A101220(1, 1, n-2), for n > 1.

Related to sum of Fibonacci(kn) over n. Cf. A099919, A058038, A138134, A053606.

Subsequence of A226538.

Sequence in context: A126348 A006731 A222036 * A179111 A093607 A005182

Adjacent sequences:  A000068 A000069 A000070 * A000072 A000073 A000074

KEYWORD

nonn,easy,nice,hear

AUTHOR

N. J. A. Sloane

EXTENSIONS

Removed attribute "conjectured" from Simon Plouffe g.f. - R. J. Mathar, Mar 11 2009

Edited by N. J. A. Sloane, Apr 04 2011

STATUS

approved

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Last modified September 1 09:40 EDT 2014. Contains 246289 sequences.