1,4

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

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. - A. O. Munagi (amunagi(AT)yahoo.com), 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 (cm(AT)caseymongoven.com), Mar 19 2006

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

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.

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

S. Burckel, Efficient methods for three strand braids (submitted).

F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 2006.

E. Deutsch, Math. Magazine, vol. 74, No. 5, 2001, p. 404, problem Q915.

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

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463

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

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

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

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

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.

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 384

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.

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

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

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) ]

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

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

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

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

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

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

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

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)

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

A000071:=1/(z-1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

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

Cf. A054761.

Antidiagonal sums of array A004070.

Right-hand column 2 of triangle A011794.

Cf. A105488, A105489.

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

Cf. A119282, A001654, A005968, A005969, A098531, A098532, A098533, A128697.

Adjacent sequences: A000068 A000069 A000070 this_sequence A000072 A000073 A000074

Sequence in context: A014968 A126348 A006731 this_sequence A093607 A005182 A094925

nonn,easy,nice

njas