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A001950
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Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.
(Formerly M1332 N0509)
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128
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2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 143, 146, 149, 151, 154, 157
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OFFSET
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1,1
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COMMENTS
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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, Nov 15 2003
A000201 and this sequence may be 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). - Philippe Deléham, Feb 20 2004
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
a(n) = Min(m: A134409(m) = A006336(n)). - Reinhard Zumkeller, Oct 24 2007
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, Nov 14 2007
Apart from the initial 0 in A090909, is this the same as that sequence? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
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REFERENCES
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C. Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324, Problem 2.
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
Fraenkel, Aviezri S., Complementary iterated floor words and the Flora game. SIAM J. Discrete Math. 24 (2010), no. 2, 570-588. - N. J. A. Sloane, May 06 2011
M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman, 1989; see p. 107.
M. Griffiths, The Golden String, Zeckendorf Representations, and the Sum of a Series, Amer. Math. Monthly, 118 (2011), 497-507.
D. J. Newman, Problem 5252, Amer. Math. Monthly, 72 (1965), 1144-1145.
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).
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 2 (1959), 190-197.
A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=1).
A. S. Fraenkel, Ratwyt, December 28 2011.
C. Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Vincent Russo and Loren Schwiebert, Beatty Sequences, Fibonacci Numbers, and the Golden Ratio, The Fibonacci Quarterly, Vol 49, Number 2, May 2011.
Eric Weisstein's World of Mathematics, Beatty Sequence, MathWorld.
Eric Weisstein's World of Mathematics, Golden ratio, MathWorld.
Eric Weisstein's World of Mathematics, Wythoff's Game, MathWorld.
Eric Weisstein's World of Mathematics, Wythoff Array
Index entries for sequences related to Beatty sequences
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FORMULA
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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
a(n) = n + floor(n*phi) - Corrected by Paul Weisenhorn.
Append a 0 to the Zeckendorf expansion (cf. A035517) of n-th term of A000201.
a(n) = A003622(n) + 1 . - Philippe Deléham, Apr 30 2004
a(n) = A000201(n) + n . - Philippe Deléham, May 02 2004
a(n) = n + floor(n*phi) - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), May 09 2007
From Paul Weisenhorn, Aug 18 2012: (Start)
a'(n) denotes the ordered complement (in N) of a(n)
a(n) = floor(n*phi^2); a'(n) = floor(n*phi) = A000201(n);
a(Fib(r-2)+j) = Fib(r)+a(j) for 0<j<=Fib(r-2); 3<r;
a'(Fib(r-1)+j) = Fib(r)+a'(j) for 0<j<=Fib(r-2); 2<r;
a(1)=2; a'(1)=1; (End)
From Paul Weisenhorn, Aug 21 2012: (Start)
with a(1)=2; a(2)=5; a'(1)=1; a'(2)=3 and 1<k and
a(k-1)<n<=a(k) one get a(n)=3*n-k; a'(n)=2*n-k; (End)
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EXAMPLE
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From Paul Weisenhorn, Aug 18 2012: (Start)
a(14)=floor(14*phi^2)=36; a'(14)=floor(14*phi)=22;
with r=9 and j=1: a(13+1)=34+2=36;
with r=8 and j=1: a'(13+1)=21+1=22; (End)
From Paul Weisenhorn, Aug 21 2012: (Start)
k=6 and a(5)=13 < n <= a(6)=15
a(14)=3*14-6=36; a'(14)=2*14-6=22;
a(15)=3*15-6=39; a'(15)=2*15-6=24; (End)
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MATHEMATICA
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Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]
Array[ Floor[ #*GoldenRatio^2] &, 60] - Robert G. Wilson v, Apr 17 2010
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PROG
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(PARI) a(n)=floor(n*(sqrt(5)+3)/2)
(Haskell)
a001950 n = a000201 n + n -- Reinhard Zumkeller, Mar 10 2013
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CROSSREFS
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a(n) = greatest k such that s(k) = n, where s = A026242. Complement of A000201.
A002251 maps between A000201 and A001950, in that A002251(A000201(n)) = A001950(n), A002251(A001950(n)) = A000201(n).
Cf. A026352, A004976, A004919.
Sequence in context: A018717 A188036 A090909 * A022841 A047480 A038127
Adjacent sequences: A001947 A001948 A001949 * A001951 A001952 A001953
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Corrected by Michael Somos, Jun 07 2000.
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STATUS
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approved
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