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 A190427 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,2,1) and []=floor. 32
 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Write a(n) = [(b*n+c)*r] - b*[n*r] - [c*r].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers. Examples: (golden ratio,2,0):  A078588, A005653, A005652 (golden ratio,2,1):  A190427 - A190430 (golden ratio,3,0):  A140397 - A190400 (golden ratio,3,1):  A140431 - A190435 (golden ratio,3,2):  A140436 - A190439 REFERENCES Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176.  Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62. LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 FORMULA a(n) = [(2*n+1)*r] - 2*[n*r] - 1, where r=(1+sqrt(5))/2. EXAMPLE a(1)=[3r]-2[r]-1=4-3-1=1. a(2)=[5r]-2[2r]-1=8-6-1=1. a(3)=[7r]-2[3r]-1=11-8-1=2. MATHEMATICA r = GoldenRatio; b = 2; c = 1; f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r]; t = Table[f[n], {n, 1, 320}] (* A190427 *) Flatten[Position[t, 0]] (* A190428 *) Flatten[Position[t, 1]] (* A190429 *) Flatten[Position[t, 2]] (* A190430 *) Table[Floor[(2*n+1)*GoldenRatio] - 2*Floor[n*GoldenRatio] -1, {n, 1, 100}] (* G. C. Greubel, Apr 06 2018 *) PROG (Python) from mpmath import mp, phi from sympy import floor mp.dps=100 def a(n): return floor((2*n + 1)*phi) - 2*floor(n*phi) - 1 print [a(n) for n in xrange(1, 132)] # Indranil Ghosh, Jul 02 2017 (PARI) for(n=1, 100, print1(floor((2*n+1)*(1+sqrt(5))/2) - 2*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ G. C. Greubel, Apr 06 2018 (MAGMA) [Floor((2*n+1)*(1+Sqrt(5))/2) - 2*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // G. C. Greubel, Apr 06 2018 CROSSREFS Cf. A078588, A190428, A190429, A190430. Sequence in context: A207869 A130210 A236459 * A287108 A287360 A035443 Adjacent sequences:  A190424 A190425 A190426 * A190428 A190429 A190430 KEYWORD nonn AUTHOR Clark Kimberling, May 10 2011 STATUS approved

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Last modified October 21 11:15 EDT 2019. Contains 328294 sequences. (Running on oeis4.)