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%I A000071 M1056 N0397
%S A000071 0,0,1,2,4,7,12,20,33,54,88,143,232,376,609,986,1596,2583,4180,6764,
%T A000071 10945,17710,28656,46367,75024,121392,196417,317810,514228,832039,
%U A000071 1346268,2178308,3524577,5702886,9227464,14930351,24157816,39088168
%N A000071 Fibonacci numbers - 1.
%C A000071 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 (deutsch(AT)duke.poly.edu), 
               Jun 04 2003
%C A000071 Number of 001-avoiding binary words of length n-3.
%C A000071 Also, sum of first n Fibonacci numbers. - Giorgi Dalakishvili (mcnamara_gio(AT)yahoo.com), 
               Apr 02 2005
%C A000071 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. - A. O. Munagi (amunagi(AT)yahoo.com), Apr 11 2005
%C A000071 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 (cm(AT)caseymongoven.com), Mar 19 2006
%C A000071 Beginning with a(2), the 'Recaman transform' (see A005132) of the Fibonacci 
               numbers (A000045). - Nick Hobson (nickh(AT)qbyte.org), Mar 01 2007
%C A000071 a(n)=2*F(n-1)+a(n-4) e.g. a(10)=0+1+1+2+3+5+8+13+21+34=88 a(6)=0+1+1+2+3+5+8=20 
               F(9)=34, so 2*F(n)+a(6)=88 where F(n)=A000045(n) (this is not quite 
               right - I can't seem to juggle the various offsets in the sequences) 
               [From J. Perry (johnandruth(AT)jrperry.orangehome.co.uk), Oct 02 
               2008]
%C A000071 Starting with nonzero terms = row sums of triangle A158950. [From Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]
%D A000071 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000071 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000071 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.
%D A000071 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 1.
%D A000071 S. Burckel, Efficient methods for three strand braids (submitted).
%D A000071 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 
               2006.
%D A000071 E. Deutsch, Math. Magazine, vol. 74, No. 5, 2001, p. 404, problem Q915.
%D A000071 R. Lagrange, Quelques re'sultats dans la me'trique des permutations, 
               Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure, Paris, 
               79 (1962), 199-241.
%D A000071 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 
               292-298.
%D A000071 A. O. Munagi, Set Partitions with Successions and Separations, Int. J. 
               Math and Math. Sc. 2005, no. 3 (2005), 451-463
%D A000071 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               155.
%D A000071 P. Xu, Growth of positive braids semigroups, Journal of Pure and Applied 
               Algebra, 1992.
%D A000071 J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 
               31 (1991), 21-29.
%H A000071 Christian G. Bower, <a href="b000071.txt">Table of n, a(n) for n=1..500</
               a>
%H A000071 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000071 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000071 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000071 A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0204320">
               Counting occurrences of some subword patterns</a>.
%H A000071 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000071 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=384">
               Encyclopedia of Combinatorial Structures 384</a>
%H A000071 R. Lagrange, <a href="http://archive.numdam.org/article/ASENS_1962_3_79_3_199_0.pdf">
               Quelques re'sultats dans la me'trique des permutations</a>, Annales 
               Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure, Paris, 79 (1962), 
               199-241.
%H A000071 A. O. Munagi, <a href="http://www.emis.de/journals/HOA/IJMMS/2005/3451.pdf">
               Set Partitions with Successions and Separations</a>,IJMMS 2005:3 
               (2005), 451-463.
%F A000071 a(0)=0, a(1)=0, a(n)=a(n-1)+a(n-2)+1.
%F A000071 Partial sum of Fibonacci numbers, G.f.: x^3/((1-x-x^2)*(1-x)) (with a(0) 
               := 0) [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) 
               ]
%F A000071 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 (ralf(AT)ark.in-berlin.de), Mar 02 2003
%F A000071 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 (benoit7848c(AT)orange.fr), 
               May 06 2003
%F A000071 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 (Gerald.McGarvey(AT)comcast.net), 
               Jul 22 2004
%F A000071 a(n)=sum{k=0..floor((n-2)/2), binomial(n-k-2, k+1)} - Paul Barry (pbarry(AT)wit.ie), 
               Sep 23 2004
%F A000071 a(n+3)=sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*2^(n-3k)} - Paul 
               Barry (pbarry(AT)wit.ie), Oct 20 2004
%F A000071 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, ... - A. O. Munagi (amunagi(AT)yahoo.com), 
               Apr 11 2005
%F A000071 a(n) = Sum[k*Fibonacci(n-k-3),{k,0,n-2}] - Ross La Haye (rlahaye(AT)new.rr.com), 
               May 31 2006
%F A000071 a(n) = term (3,2) in the 3x3 matrix [1,1,0; 1,0,0; 1,0,1]^n. - Alois 
               P. Heinz (heinz(AT)hs-heilbronn.de), Jul 24 2008
%p A000071 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)
%p A000071 with(combinat): a:=n->(sum((fibonacci(j)),j=0..n)): seq(a(n),n=-1..36); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
%p A000071 A000071:=1/(z-1)/(z^2+z-1); [S. Plouffe in his 1992 dissertation, dropping 
               initial zeros.]
%p A000071 a := n -> (Matrix ([[1,1,0], [1,0,0], [1,0,1]])^n)[3,2]; seq (a(n), n=0..50); 
               - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 24 2008
%t A000071 Table[f=Fibonacci[k];f-1,{k,1,40,1}] (Vladimir Orlovsky, Jul 21 2008)
%t A000071 Table[Sum[Fibonacci[i], {i, 0, n}], {n, -1, 36}] [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
%o A000071 (PARI) a(n)=if(n<1,0,fibonacci(n)-1)
%o A000071 sage: from sage.combinat.sloane_functions import recur_gen2 sage: it 
               = recur_gen2(1,1,1,1) sage: [it.next()-1 for i in xrange(0,39)] - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 06 2008
%Y A000071 Cf. A054761.
%Y A000071 Antidiagonal sums of array A004070.
%Y A000071 Right-hand column 2 of triangle A011794.
%Y A000071 Cf. A105488, A105489.
%Y A000071 a(n) = A101220(1, 1, n-2), for n > 1.
%Y A000071 Cf. A119282, A001654, A005968, A005969, A098531, A098532, A098533, A128697.
%Y A000071 A158950 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]
%Y A000071 Sequence in context: A014968 A126348 A006731 this_sequence A093607 A005182 
               A094925
%Y A000071 Adjacent sequences: A000068 A000069 A000070 this_sequence A000072 A000073 
               A000074
%K A000071 nonn,easy,nice
%O A000071 1,4
%A A000071 N. J. A. Sloane (njas(AT)research.att.com).
%E A000071 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009

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