%I M1056 N0397 #423 Nov 07 2024 16:32:41

%S 0,0,1,2,4,7,12,20,33,54,88,143,232,376,609,986,1596,2583,4180,6764,

%T 10945,17710,28656,46367,75024,121392,196417,317810,514228,832039,

%U 1346268,2178308,3524577,5702886,9227464,14930351,24157816,39088168,63245985,102334154

%N a(n) = Fibonacci(n) - 1.

%C 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

%C 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 [For a(5) = 4 we have 2143, 1324, 2134 and 1243. - _Jon Perry_, Sep 14 2013]

%C Number of 001-avoiding binary words of length n-3. a(n) is the 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

%C 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

%C Beginning with a(2), the "Recamán transform" (see A005132) of the Fibonacci numbers (A000045). - _Nick Hobson_, Mar 01 2007

%C Starting with nonzero terms, a(n) gives the row sums of triangle A158950. - _Gary W. Adamson_, Mar 31 2009

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

%C 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

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

%C a(n+1) is the number of compositions of n with maximal part 2. - _Joerg Arndt_, May 21 2013

%C 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 if and only if they have disjoint grandparents and same great-grandparent. Consequence: a(n) = 2*a(n-1) - a(n-3). - _Hermann Stamm-Wilbrandt_, Jul 06 2014

%C 2 and 7 are the only prime numbers in this sequence. - _Emmanuel Vantieghem_, Oct 01 2014

%C From _Russell Jay Hendel_, Mar 15 2015: (Start)

%C We can establish _Gerald McGarvey_'s conjecture mentioned in the Formula section, however we require n > 4. We need the following 4 prerequisites.

%C (1) a(n) = F(n) - 1, with {F(n)}_{n >= 1} the Fibonacci numbers A000045. (2) (Binet form) F(n) = (d^n - e^n)/sqrt(5) with d = phi and e = 1 - phi, de = -1 and d + e = 1. It follows that a(n) = (d(n) - e(n))/sqrt(5) - 1. (3) To prove floor(x) = y is equivalent to proving that x - y lies in the half-open interval [0, 1). (4) The series {s(n) = c1 x^n + c2}_{n >= 1}, with -1 < x < 0, and c1 and c2 positive constants, converges by oscillation with s(1) < s(3) < s(5) < ... < s(6) < s(4) < s(2). If follows that for any odd n, the open interval (s(n), s(n+1)) contains the subsequence {s(t)}_{t >= n + 2}. Using these prerequisites we can analyze the conjecture.

%C Using prerequisites (2) and (3) we see we must prove, for all n > 4, that d((d^(n-1) - e^(n-1))/sqrt(5) - 1) - (d^n - e^n)/sqrt(5) + 1 + c lies in the interval [0, 1). But de = -1, implying de^(n-1) = -e^(n-2). It follows that we must equivalently prove (for all n > 4) that E(n, c) = (e^(n-2) + e^n)/sqrt(5) + 1 - d + c = e^(n-2) (e^2 + 1)/sqrt(5) + e + c lies in [0, 1). Clearly, for any particular n, E(n, c) has extrema (maxima, minima) when c = 2*(1-d) and c = (1+d)*(1-d). Therefore, the proof is completed by using prerequisite (4). It suffices to verify E(5, 2*(1-d)) = 0, E(6, 2*(1-d)) = 0.236068, E(5, (1-d)*(1+d)) = 0.618034, E(6, (1-d)*(1+d)) = 0.854102, all lie in [0, 1).

%C (End)

%C a(n) can be shown to be the number of distinct nonempty matchings on a path with n vertices. (A matching is a collection of disjoint edges.) - _Andrew Penland_, Feb 14 2017

%C Also, for n > 3, the lexicographically earliest sequence of positive integers such that {phi*a(n)} is located strictly between {phi*a(n-1)} and {phi*a(n-2)}. - _Ivan Neretin_, Mar 23 2017

%C From _Eric M. Schmidt_, Jul 17 2017: (Start)

%C Number of sequences (e(1), ..., e(n-2)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) <= e(k). [Martinez and Savage, 2.5]

%C Number of sequences (e(1), ..., e(n-2)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) <= e(k) and e(i) != e(k). [Martinez and Savage, 2.5]

%C (End)

%C Numbers whose Zeckendorf (A014417) and dual Zeckendorf (A104326) representations are the same: alternating digits of 1 and 0. - _Amiram Eldar_, Nov 01 2019

%C a(n+2) is the length of the longest array whose local maximum element can be found in at most n reveals. See link to the puzzle by Alexander S. Kulikov. - _Dmitry Kamenetsky_, Aug 08 2020

%C a(n+2) is the number of nonempty subsets of {1,2,...,n} that contain no consecutive elements. For example, the a(6)=7 subsets of {1,2,3,4} are {1}, {2}, {3}, {4}, {1,3}, {1,4} and {2,4}. - _Muge Olucoglu_, Mar 21 2021

%C a(n+3) is the number of allowed patterns of length n in the even shift (that is, a(n+3) is the number of binary words of length n in which there are an even number of 0s between any two occurrences of 1). For example, a(7)=12 and the 12 allowed patterns of length 4 in the even shift are 0000, 0001, 0010, 0011, 0100, 0110, 0111, 1000, 1001, 1100, 1110, 1111. - _Zoran Sunic_, Apr 06 2022

%C Conjecture: for k a positive odd integer, the sequence {a(k^n): n >= 1} is a strong divisibility sequence; that is, for n, m >= 1, gcd(a(k^n), a(k^m)) = a(k^gcd(n,m)). - _Peter Bala_, Dec 05 2022

%C In general, the sum of a second-order linear recurrence having signature (c,d) will be a third-order recurrence having a signature (c+1,d-c,-d). - _Gary Detlefs_, Jan 05 2023

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

%D GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 28.

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

%D D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

%H Christian G. Bower, <a href="/A000071/b000071.txt">Table of n, a(n) for n = 1..500</a>

%H Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, <a href="https://arxiv.org/abs/2006.13002">From Unequal Chance to a Coin Game Dance: Variants of Penney's Game</a>, arXiv:2006.13002 [math.HO], 2020.

%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2009.02669">Symbolic dynamical scales: modes, orbitals, and transversals</a>, arXiv:2009.02669 [math.DS], 2020.

%H Kassie Archer and Aaron Geary, <a href="https://arxiv.org/abs/2312.14351">Powers of permutations that avoid chains of patterns</a>, arXiv:2312.14351 [math.CO], 2023. See p. 15.

%H Mohammad K. Azarian, <a href="http://cs.ucmo.edu/~mjms/1990.2/azar/azar.pdf">The Generating Function for the Fibonacci Sequence</a>, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.

%H Mohammad K. Azarian, <a href="http://cs.ucmo.edu/~mjms/2004.1/azar6/Febmjmsazar6.pdf">A Generalization of the Climbing Stairs Problem II</a>, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.

%H J.-L. Baril and J.-M. Pallo, <a href="http://jl.baril.u-bourgogne.fr/Motzkin.pdf">Motzkin subposet and Motzkin geodesics in Tamari lattices</a>, 2013.

%H Andrew M. Baxter and Lara K. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf">Ascent sequences avoiding pairs of patterns</a>, 2014.

%H Serge Burckel, <a href="http://dx.doi.org/10.1006/jsco.2000.0473">Syntactical methods for braids of three strands</a>, J. Symbolic Comput. 31 (2001), no. 5, 557-564.

%H Alexander Burstein and Toufik Mansour, <a href="https://arxiv.org/abs/math/0204320">Counting occurrences of some subword patterns</a>, arXiv:math/0204320 [math.CO], 2002-2003.

%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H Fan Chung and R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194.

%H Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, <a href="http://arxiv.org/abs/1606.06228">Hyperfibonacci Sequences and Polytopic Numbers</a>, arXiv:1606.06228 [math.CO], 2016.

%H Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p22">Non-contiguous pattern avoidance in binary trees</a>. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013

%H Emeric Deutsch, <a href="http://www.jstor.org/stable/2691040">Problem Q915</a>, Math. Magazine, vol. 74, No. 5, 2001, p. 404.

%H Christian Ennis, William Holland, Omer Mujawar, Aadit Narayanan, Frank Neubrander, Marie Neubrander, and Christina Simino, <a href="https://arxiv.org/abs/2107.01029">Words in Random Binary Sequences I</a>, arXiv:2107.01029 [math.GM], 2021.

%H Fumio Hazama, <a href="http://dx.doi.org/10.1016/j.disc.2011.06.008">Spectra of graphs attached to the space of melodies</a>, Discrete Math., 311 (2011), 2368-2383. See Table 2.1.

%H Yasuichi Horibe, <a href="http://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [From _Emeric Deutsch_, Jun 14 2010]

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=384">Encyclopedia of Combinatorial Structures 384</a>

%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 96.

%H Scott O. Jones and P. Mark Kayll, <a href="https://combinatorialpress.com/jcmcc-articles/volume-057/constructing-edge-labellings-of-k_n-with-constant-length-hamilton-cycles/">Constructing Edge-labellings of K_n, with Constant-length Hamilton Cycles</a>, J. Comb. Math. Comb. Comp. (2006) Vol. 57, pp. 83-95. See p. 92.

%H Tamara Kogan, L. Sapir, A. Sapir, and A. Sapir, <a href="https://doi.org/10.1016/j.apnum.2016.08.012">The Fibonacci family of iterative processes for solving nonlinear equations</a>, Applied Numerical Mathematics 110 (2016) 148-158.

%H Alexander S. Kulikov, <a href="https://puzzling.stackexchange.com/questions/100890/find-local-maximum-in-an-integer-sequence">Find Local Maximum in an Integer Sequence</a>, Puzzling Stack Exchange, 2020.

%H René Lagrange, <a href="http://archive.numdam.org/article/ASENS_1962_3_79_3_199_0.pdf">Quelques résultats dans la métrique des permutations</a>, Annales Scientifiques de l'Ecole Normale Supérieure, Paris, 79 (1962), 199-241.

%H D. A. Lind, <a href="http://www.fq.math.ca/Scanned/3-4/lind.pdf">On a class of nonlinear binomial sums</a>, Fib. Quart., 3 (1965), 292-298.

%H Rui Liu and Feng-Zhen Zhao, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Liu/liu10.html">On the Sums of Reciprocal Hyperfibonacci Numbers and Hyperlucas Numbers</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.4.5. - From _N. J. A. Sloane_, Oct 05 2012

%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.

%H Augustine O. Munagi, <a href="http://dx.doi.org/10.1155/IJMMS.2005.451">Set Partitions with Successions and Separations</a>,IJMMS 2005:3 (2005), 451-463.

%H Sam Northshield, <a href="http://www.jstor.org/stable/10.4169/amermathmont.117.7.581">Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...</a>, Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/notredame.pdf">Pattern avoidance in trees</a>, (slides from a talk, mentions many sequences), 2012.

%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/ascseq.pdf">Pattern-avoiding ascent sequences</a>, Slides from a talk, 2015.

%H Stacey Wagner, <a href="http://via.library.depaul.edu/depaul-disc/vol2/iss1/2">Enumerating Alternating Permutations with One Alternating Descent</a>, DePaul Discoveries: Vol. 2: Iss. 1, Article 2.

%H Hsin-Po Wang and Chi-Wei Chin, <a href="https://arxiv.org/abs/2405.17499">On Counting Subsequences and Higher-Order Fibonacci Numbers</a>, arXiv:2405.17499 [cs.IT], 2024. See p. 2.

%H Arthur T. White, <a href="http://dx.doi.org/10.1017/S0305004100061053">Ringing the changes</a>, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.

%H Peijun Xu, <a href="http://dx.doi.org/10.1016/0022-4049(92)90078-T">Growth of positive braids semigroups</a>, Journal of Pure and Applied Algebra, 1992.

%H J. L. Yucas, <a href="/A006980/a006980.pdf">Counting special sets of binary Lyndon words</a>, Ars Combin., 31 (1991), 21-29. (Annotated scanned copy)

%H Jianqiang Zhao, <a href="http://arxiv.org/abs/1412.8044">Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras</a>, arXiv preprint arXiv:1412.8044 [math.NT], 2014. See Table 9, line 1.

%H Li-Na Zheng, Rui Liu, and Feng-Zhen Zhao, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Zheng/zheng5.html">On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers</a>, Journal of Integer Sequences, Vol. 17 (2014), #14.1.4.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).

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

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

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

%F G.f.: x^3/((1-x-x^2)*(1-x)). - _Simon Plouffe_ in his 1992 dissertation, dropping initial 0's

%F a(n) = 2*a(n-1) - a(n-3). - _R. H. Hardin_, Apr 02 2011

%F Partial sums of Fibonacci numbers. - _Wolfdieter Lang_

%F 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

%F 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

%F 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 > 4. - _Gerald McGarvey_, Jul 22 2004. This is true provided n > 3 is changed to n > 4, see proof in Comments section. - _Russell Jay Hendel_, Mar 15 2015

%F a(n) = Sum_{k = 0..floor((n-2)/2)} binomial(n-k-2, k+1). - _Paul Barry_, Sep 23 2004

%F a(n+3) = Sum_{k = 0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*2^(n-3*k). - _Paul Barry_, Oct 20 2004

%F 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

%F a(n) = Sum_{k = 0..n-2} k*Fibonacci(n - k - 3). - _Ross La Haye_, May 31 2006

%F a(n) = term (3, 2) in the 3 X 3 matrix [1, 1, 0; 1, 0, 0; 1, 0, 1]^(n-1). - _Alois P. Heinz_, Jul 24 2008

%F For n >= 4, a(n) = ceiling(phi*a(n-1)), where phi is the golden ratio. - _Vladimir Shevelev_, Jul 04 2010

%F 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 0's 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

%F A000119(a(n)) = 1. - _Reinhard Zumkeller_, Dec 28 2012

%F a(n) = A228074(n - 1, 2) for n > 2. - _Reinhard Zumkeller_, Aug 15 2013

%F 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

%F A083368(a(n+3)) = n. - _Reinhard Zumkeller_, Aug 10 2014

%F E.g.f.: 1 - exp(x) + 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - _Ilya Gutkovskiy_, Jun 15 2016

%F a(n) = A000032(3+n) - 1 mod A000045(3+n). - _Mario C. Enriquez_, Apr 01 2017

%F a(n) = Sum_{i=0..n-2} Fibonacci(i). - Giorgi Dalakishvili (mcnamara_gio(AT)yahoo.com), Apr 02 2005 [corrected by _Doug Bell_, Jun 01 2017]

%F a(n+2) = Sum_{j = 0..floor(n/2)} Sum_{k = 0..j} binomial(n - 2*j, k+1)*binomial(j, k). - _Tony Foster III_, Sep 08 2017

%F From _Peter Bala_, Nov 12 2021: (Start)

%F a(4*n) = Fibonacci(2*n+1)*Lucas(2*n-1) = A081006(n);

%F a(4*n+1) = Fibonacci(2*n)*Lucas(2*n+1) = A081007(n);

%F a(4*n+2) = Fibonacci(2*n)*Lucas(2*n+2) = A081008(n);

%F a(4*n+3) = Fibonacci(2*n+2)*Lucas(2*n+1) = A081009(n). (End)

%F G.f.: x^3/((1 - x - x^2)*(1 - x)) = Sum_{n >= 0} (-1)^n * x^(n+3) *( Product_{k = 1..n} (k - x)/Product_{k = 1..n+2} (1 - k*x) ) (a telescoping series). - _Peter Bala_, May 08 2024

%p A000071 := proc(n) combinat[fibonacci](n)-1 ; end proc; # _R. J. Mathar_, Apr 07 2011

%p 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

%t Fibonacci[Range[40]] - 1 (* or *) LinearRecurrence[{2, 0, -1}, {0, 0, 1}, 40] (* _Harvey P. Dale_, Aug 23 2013 *)

%t Join[{0}, Accumulate[Fibonacci[Range[0, 39]]]] (* _Alonso del Arte_, Oct 22 2017, based on Giorgi Dalakishvili's formula *)

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

%o (Magma) [Fibonacci(n)-1: n in [1..60]]; // _Vincenzo Librandi_, Apr 04 2011

%o (Haskell)

%o a000071 n = a000071_list !! n

%o a000071_list = map (subtract 1) $ tail a000045_list

%o -- _Reinhard Zumkeller_, May 23 2013

%o (SageMath) [fibonacci(n)-1 for n in range(1,60)] # _G. C. Greubel_, Oct 21 2024

%Y Cf. A000032, A000045, A000119, A001611, A001654, A001911, A005968, A005969.

%Y Cf. A006327, A014417, A054761, A081006, A081007, A081008, A081009, A083368.

%Y Cf. A098531, A098532, A098533, A101220, A104326, A105488, A105489, A119282.

%Y Cf. A128697, A157725, A157726, A157727, A157728, A157729, A158950, A167616.

%Y Cf. A228074.

%Y Antidiagonal sums of array A004070.

%Y Right-hand column 2 of triangle A011794.

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

%Y Subsequence of A226538. Also a subsequence of A061489.

%K nonn,easy,nice,hear,changed

%O 1,4

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, Apr 04 2011