%I A001950 M1332 N0509
%S A001950 2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,
%T A001950 62,65,68,70,73,75,78,81,83,86,89,91,94,96,99,102,104,107,109,112,
%U A001950 115,117,120,123,125,128,130,133,136,138,141,143,146,149,151,154,157
%N A001950 Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where 
               phi = (1+sqrt(5))/2.
%C A001950 Indices at which blocks (1;0) occur in infinite Fibonacci word; i.e. 
               n such that A005614(n-2) = 0 and A005614(n-1) = 1 - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Nov 15 2003
%C A001950 A000201 and this sequence 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
%C A001950 a(n) = n-th integer which is not equal to the floor of any multiple of 
               phi, where phi = (1+sqrt(5))/2 = golden number. - Philippe LALLOUET 
               (philip.lallouet(AT)wanadoo.fr), May 09 2007
%C A001950 a(n) = Min(m: A134409(m) = A006336(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 24 2007
%C A001950 Write A for A000201 and B for the present sequence (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, the present sequence). 
               Typical complementary equations: AB=A+B (=A003623), BB=A+2B (=A101864), 
               BBB=3A+5B (=A134864). - Clark Kimberling (ck6(AT)evansville.edu), 
               Nov 14 2007
%C A001950 Apart from the initial 0 in A090909, is this the same as that sequence? 
               - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
%D A001950 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001950 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001950 C. Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324, Problem 
               2.
%D A001950 L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related 
               to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
%D A001950 I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 
               2 (1959), 190-197.
%D A001950 A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, 
               Amer. Math. Monthly, 89 (1982), 353-361 (the case a=1).
%D A001950 Clark Kimberling, Complementary Equations, Journal of Integer Sequences, 
               Vol. 10 (2007), Article 07.1.4.
%D A001950 D. J. Newman, Problem 5252, Amer. Math. Monthly, 72 (1965), 1144-1145.
%D A001950 X. Sun, Wythoff's sequence ..., Discr. Math., 300 (2005), 180-195.
%D A001950 J. C. Turner, The alpha and the omega of the Wythoff pairs, Fib. Q., 
               27 (1989), 76-86.
%D A001950 I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics 
               I, Amer Math. Soc., 2001.
%D A001950 C. Kimberling, Complementary equations and Wythoff sequences, preprint, 
               2007.
%H A001950 T. D. Noe, <a href="b001950.txt">Table of n, a(n) for n=1..1000</a>
%H A001950 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 
               3 (2000), #00.2.8.
%H A001950 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BeattySequence.html">Link to a section of The World of Mathematics.</
               a>
%H A001950 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GoldenRatio.html">Link to a section of The World of Mathematics.</
               a>
%H A001950 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WythoffsGame.html">Link to a section of The World of Mathematics.</
               a>
%H A001950 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WythoffArray.html">Wythoff Array</a>
%H A001950 <a href="Sindx_Be.html#Beatty">Index entries for sequences related to 
               Beatty sequences</a>
%F A001950 a(n) = n + floor(2 n phi). In general b(n) = floor(n*phi^m) = Fibonacci(m-1)*n 
               + floor(Fibonacci(m)*n*phi). - Benoit Cloitre, Mar 18, 2003
%F A001950 Append a 0 to the Zeckendorf expansion (cf. A035517) of n-th term of 
               A000201.
%F A001950 a(n) = A003622(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Apr 30 2004
%F A001950 a(n) = A000201(n) + n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               May 02 2004
%F A001950 a(n) = n + floor(n*phi) - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), 
               May 09 2007
%t A001950 Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]
%o A001950 (PARI) a(n)=floor(n*(sqrt(5)+3)/2)
%Y A001950 a(n) = greatest k such that s(k) = n, where s = A026242. Complement of 
               A000201.
%Y A001950 Cf. A004976, A004919.
%Y A001950 A002251 maps between A000201 and A001950, in that A002251(A000201(n)) 
               = A001950(n), A002251(A001950(n)) = A000201(n).
%Y A001950 Cf. A026352.
%Y A001950 Sequence in context: A026340 A018717 A090909 this_sequence A022841 A047480 
               A038127
%Y A001950 Adjacent sequences: A001947 A001948 A001949 this_sequence A001951 A001952 
               A001953
%K A001950 nonn,easy,nice
%O A001950 1,1
%A A001950 N. J. A. Sloane (njas(AT)research.att.com).
%E A001950 Corrected by Michael Somos, Jun 07 2000.