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 A003623 Wythoff AB-numbers: [[n*phi^2]*phi], where phi = (1+sqrt(5))/2. (Formerly M2715) 23
 3, 8, 11, 16, 21, 24, 29, 32, 37, 42, 45, 50, 55, 58, 63, 66, 71, 76, 79, 84, 87, 92, 97, 100, 105, 110, 113, 118, 121, 126, 131, 134, 139, 144, 147, 152, 155, 160, 165, 168, 173, 176, 181, 186, 189, 194, 199, 202, 207, 210, 215, 220, 223, 228, 231, 236, 241, 244, 249 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Previous name was: "From a 3-way splitting of positive integers: [[n*phi^2]*phi]." Union of A001950 & A003622 & A003623 = A000027. a(n) is odd if and only if n is odd. [Clark Kimberling, Apr 21 2011] A005614(a(n)-1)=1 and A005614(a(n))=1, n>=1. Because Wythoff AB-numbers (see the formula section) mark the first entry of pairs of 1s in the rabbit sequence A005614(n-1), n>=1. - Wolfdieter Lang, Jun 28 2011 a(n) = k if and only if A270788(k) = 3, where A270788 is the infinite Fibonacci word on {1,2,3}. - Michel Dekking, Sep 07 2016 REFERENCES J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 10. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Nathaniel Johnston, Table of n, a(n) for n = 1..10000 J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018. Aviezri S. Fraenkel, Complementary iterated floor words and the Flora game, SIAM J. Discrete Math. 24 (2010), no. 2, 570-588. - From N. J. A. Sloane, May 06 2011 A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018. C. Kimberling, Complementary equations and Wythoff Sequences, JIS 11 (2008) 08.3.3 C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273. U. Larsson, N. Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4. Aviezri S. Fraenkel, The Raleigh game, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1. F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015 (see page 2, essential numbers). FORMULA a(n) = [n*phi] + [n*phi^2] = A000201(n) + A001950(n). a(n) = 2*[n*phi]+n. a(n) = A(B(n)) with A(k):=A000201(k) and B(k):=A001950(k), k>=1 (Wythoff AB-numbers). MAPLE A003623:=proc(n) return floor(floor(n*(3+sqrt(5))/2)*(1+sqrt(5))/2); end:seq(A003623(n), n=1..59); # Nathaniel Johnston, Apr 21 2011 MATHEMATICA f[n_] := Floor[ GoldenRatio * Floor[ n * GoldenRatio^2]]; Array[f, 47] (* another *) Table[n+2Floor[n*GoldenRatio], {n, 1, 100}] PROG (Python) from sympy import floor from mpmath import phi def a(n): return floor(n*phi) + floor(n*phi**2) # Indranil Ghosh, Jun 10 2017 CROSSREFS Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. Sequence in context: A047470 A184401 A190251 * A190463 A190435 A188032 Adjacent sequences:  A003620 A003621 A003622 * A003624 A003625 A003626 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Name improved by Michel Dekking, Sep 07 2016 STATUS approved

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Last modified October 20 15:15 EDT 2019. Contains 328267 sequences. (Running on oeis4.)