1,2

This is the unique sequence a satisfying a'(n)=a(a(n))+1 for all n in the set N of natural numbers, where a' denotes the ordered complement (in N) of a. - Clark Kimberling (ck6(AT)evansville.edu), Feb 17 2003

This sequence and A001950 may defined as follows . Consider the maps a -> ab, b -> a, starting from a(1) = a; then A000201 gives the indices of a, A001950 gives the indices of b . The sequence of letters in the infinite word begins a, b, a, a, b, a, b, a, a, b, a, ... Setting a = 0, b = 1 gives A003849 (offset 0); setting a = 1, b = 0 gives A005614 (offset 0) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004

These are the numbers whose lazy Fibonacci representation (see A095791) includes 1; the complementary sequence (the upper Wythoff sequence, A001950) are the numbers whose lazy Fibonacci representation includes 2 but not 1.

a(n) is the unique monotonic sequence satisfying a(1)=1 and the condition "if n is in the sequence then n+(rank of n) is not in the sequence" (e.g. a(4)=6 so 6+4=10 and 10 is not in the sequence) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 31 2006

Write A for A000201 and B for A001950 (the upper Wythoff sequence, complement of A). Then the composite sequences AA, AB, BA, BB, AAA, AAB,...,BBB,... appear in many complementary equations having solution A000201 (or equivalently, A001950). Typical complementary equations: AB=A+B (=A003623), BB=A+2B (=A101864), BBB=3A+5B (=A134864). - Clark Kimberling (ck6(AT)evansville.edu), Nov 14 2007

Cumulative sum of A001468 terms. [From Eric Angelini (eric.angelini(AT)skynet.be), Aug 19 2008]

Shiri Artstein-avidan, Aviezri S. Fraenkel and Vera T. Sos, A two-parameter family of an extension of Beatty sequences, http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/coatp8.pdf, Discrete Math., 308 (2008), 4578-4588.

M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 2 (1959), 190-197.

P. J. Downey and R. E. Griswold, On a family of nested recurrences, Fib. Quart., 22 (1984), 310-317.

A. S. Fraenkel, The bracket function and complementary sets of integers, Canad. J. Math., 21 (1969), 6-27. [History, references, generalization]

A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=1).

David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.

A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, pp. 513-514.

H. Grossman, A set containing all integers, Amer. Math. Monthly, 69 (1962), 532-533.

C. Kimberling, Complementary equations and Wythoff sequences, preprint, 2007.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

D. J. Newman, Problem 5252, Amer. Math. Monthly, 72 (1965), 1144-1145.

Problem 3117, Amer. Math. Monthly, 34 (1927), 158-159.

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).

K. B. Stolarsky, Beatty sequences, continued fractions and certain shift operators, Canad. Math. Bull., 19 (1976), 473-482.

X. Sun, Wythoff's sequence ..., Discr. Math., 300 (2005), 180-195.

J. C. Turner, The alpha and the omega of the Wythoff pairs, Fib. Q., 27 (1989), 76-86.

I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.

N. J. A. Sloane, The first 10000 terms

Joerg Arndt, Fxtbook

J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.

E. J. Barbeau, J. Chew and S. Tanny, A matrix dynamics approach to Golomb's recusion, Electronic J. Combinatorics, #4.1 16 1997.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

C. Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Classic Sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Wythoff Array

Index entries for sequences related to Beatty sequences

Index entries for sequences of the a(a(n)) = 2n family

Zeckendorf expansion of n (cf. A035517) ends with an even number of 0's.

Other properties: a(1)=1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition "n is in the sequence if and only if a(n)+1 is not in sequence".

a(1) = 1; for n>0, a(n+1) = a(n)+1 if n is not in sequence, a(n+1) = a(n)+2 if n is in sequence.

a(a(n)) = floor[n*phi^2] - 1 = A003622(n).

{a(k)} union {a(k)+1} = {1, 2, 3, 4, ...}. Hence a(1)=1; for n>1, a(a(n))=a(a(n)-1)+2, a(a(n)+1)=a(a(n))+1. - Benoit Cloitre, Mar 08, 2003

{a(n)} is a solution to the recurrence a(a(n)+n) = 2*a(n)+n, a(1)=1 (see Barbeau et al.).

a(n) = A001950(n) - n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 02 2004

a(0) = 0; a(n) = n + max{ k : a(k) < n}. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 11 2004

Digits := 100; t := evalf((1+sqrt(5))/2); A000201 := n->floor(t*n);

Table[Floor[N[n*(1+Sqrt[5])/2]], {n, 1, 75}]

(PARI) a(n)=floor(n*(sqrt(5)+1)/2)

a(n) = least k such that s(k) = n, where s = A026242. Complement of A001950. See also A058066.

The permutation A002251 maps between this sequence and A001950, in that A002251(a(n)) = A001950(n), A002251(A001950(n)) = a(n).

First differences give A014675. a(n) = A022342(n) + 1 = A005206(n) + n + 1. a(2n)-a(n)=A007067(n). a(a(a(n)))-a(n) = A026274(n-1). - Benoit Cloitre, Mar 08 2003.

Sequence in context: A085270 A090908 A066096 this_sequence A000202 A026339 A109259

Adjacent sequences: A000198 A000199 A000200 this_sequence A000202 A000203 A000204

nonn,easy,nice

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