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

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]

Starting with nonzero terms = row sums of triangle A158950. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]

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.

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).

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

Index entries for sequences related to linear recurrences with constant coefficients

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(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

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); [S. Plouffe in his 1992 dissertation, dropping initial zeros.]

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

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

Table[Sum[Fibonacci[i], {i, 0, n}], {n, -1, 36}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]

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

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

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.

A158950 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]

Sequence in context: A014968 A126348 A006731 this_sequence A093607 A005182 A094925

Adjacent sequences: A000068 A000069 A000070 this_sequence A000072 A000073 A000074

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009