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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 range(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 A333948 A287360

Adjacent sequences:  A190424 A190425 A190426 * A190428 A190429 A190430

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 10 2011

STATUS

approved

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Last modified June 23 09:06 EDT 2021. Contains 345395 sequences. (Running on oeis4.)