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 A003622 The Wythoff compound sequence AA: a(n) = floor(n*phi^2) - 1, where phi = (1+sqrt(5))/2. (Formerly M3278) 82
 1, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 95, 98, 101, 103, 106, 108, 111, 114, 116, 119, 122, 124, 127, 129, 132, 135, 137, 140, 142, 145, 148, 150, 153, 156, 158, 161, 163, 166 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also, integers with "odd" Zeckendorf expansions (end with ...+F_2 = ...+1) (Fibonacci-odd numbers); first column of Wythoff array A035513; from a 3-way splitting of positive integers. [Edited by Peter Munn, Sep 16 2022] Also, numbers k such that A005206(k) = A005206(k+1). Also k such that A022342(A005206(k)) = k+1 (for all other k's this is k). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001 Also, positions of 1's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011 From Amiram Eldar, Sep 03 2022: (Start) Numbers with an odd number of trailing 1's in their dual Zeckendorf representation (A104326), i.e., numbers k such that A356749(k) is odd. The asymptotic density of this sequence is 1 - 1/phi (A132338). (End) REFERENCES A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 62. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition. C. Kimberling, "Stolarsky interspersions", Ars Combinatoria 39 (1995) 129-138. D. R. Morrison, "A Stolarsky array of Wythoff pairs", in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136. J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 10. N. J. A. Sloane and Simon Plouffe, Encyclopedia of Integer Sequences, Academic Press, 1995: this sequence appears twice, as both M3277 and M3278. LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018. A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 62. Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - N. J. A. Sloane, Jun 10 2012 Aviezri S. Fraenkel, The Raleigh game, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1. Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), Article 15.11.8. V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977. Clark Kimberling, Interspersions. Clark Kimberling, Complementary equations and Wythoff Sequences, JIS 11 (2008), Article 08.3.3. Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5. Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021). Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273. L. Lindroos, A. Sills and H. Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., 52 (2014), 61-65. M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.5. Mathematics Stack Exchange, Golden ratio and floor function floor(phi^2*n) - floor(phi*floor(phi*n)) = 1. N. J. A. Sloane, Classic Sequences N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence) Jiemeng Zhang, Zhixiong Wen, and Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), Article P2.52. FORMULA a(n) = floor(n*phi) + n - 1. [Corrected by Jianing Song, Aug 18 2022] a(n) = floor(floor(n*phi)*phi) = A000201(A000201(n)). [See the Mathematics Stack Exchange link for a proof of the equivalence of the definition. - Jianing Song, Aug 18 2022] a(n) = 1 + A022342(1 + A022342(n)). G.f.: 1 - (1-x)*Sum_{n>=1} x^a(n) = 1/1 + x/1 + x^2/1 + x^3/1 + x^5/1 + x^8/1 + ... + x^F(n)/1 + ... (continued fraction where F(n)=n-th Fibonacci number). - Paul D. Hanna, Aug 16 2002 a(n) = A001950(n) - 1. - Philippe Deléham, Apr 30 2004 a(n) = A022342(n) + n. - Philippe Deléham, May 03 2004 MAPLE A003622 := proc(n) n+floor(n*(1+sqrt(5))/2)-1 ; end proc: # R. J. Mathar, Jan 25 2015 # Maple code for the Wythoff compound sequences, from N. J. A. Sloane, Mar 30 2016 # The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp. # Assume files out1, out2 contain lists of the terms in the base sequences A and B from their b-files read out1; read out2; b[0]:=b1: b[1]:=b2: w2:=(i, j, n)->b[i][b[j][n]]; w3:=(i, j, k, n)->b[i][b[j][b[k][n]]]; for i from 0 to 1 do lprint("name=", i); lprint([seq(b[i][n], n=1..100)]): od: for i from 0 to 1 do for j from 0 to 1 do lprint("name=", i, j); lprint([seq(w2(i, j, n), n=1..100)]); od: od: for i from 0 to 1 do for j from 0 to 1 do for k from 0 to 1 do lprint("name=", i, j, k); lprint([seq(w3(i, j, k, n), n=1..100)]); od: od: od: MATHEMATICA With[{c=GoldenRatio^2}, Table[Floor[n c]-1, {n, 70}]] (* Harvey P. Dale, Jun 11 2011 *) Range[70]//Floor[#*GoldenRatio^2]-1& (* Waldemar Puszkarz, Oct 10 2017 *) PROG (PARI) a(n)=floor(n*(sqrt(5)+3)/2)-1 (PARI) a(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ Michel Marcus, Sep 17 2022 (Haskell) a003622 n = a003622_list !! (n-1) a003622_list = filter ((elem 1) . a035516_row) [1..] -- Reinhard Zumkeller, Mar 10 2013 (Python) from sympy import floor from mpmath import phi def a(n): return floor(n*phi**2) - 1 # Indranil Ghosh, Jun 09 2017 (Python) from math import isqrt def A003622(n): return (n+isqrt(5*n**2)>>1)+n-1 # Chai Wah Wu, Aug 11 2022 CROSSREFS Positions of 1's in A003849. Complement of A022342. Cf. A066096, A139764, A035516, A026273, A104326, A132338, A356749. The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp. The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021 Sequence in context: A190304 A189366 A066095 * A330215 A189533 A047408 Adjacent sequences: A003619 A003620 A003621 * A003623 A003624 A003625 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane, Mira Bernstein, Marc LeBrun STATUS approved

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Last modified May 28 15:40 EDT 2023. Contains 363019 sequences. (Running on oeis4.)