A339765 a(n) = 2*floor(n*phi) - 3*n, where phi = (1+sqrt(5))/2.

-1, 0, -1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 10, 11, 12, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 15, 16, 17, 16, 17, 16

Offset: 1

Author

Primoz Pirnat, Dec 16 2020

Keywords

• sign

Comments

a(n) are coefficients in the formulas for multiplication of fractional parts of multiples of the golden mean:

(I) frac(b*phi)*frac(c*phi) = 1-frac(d*phi); d = 2*b*c+a(b)*c/2+a(c)*b/2;

(IIa) frac(b*phi)*(1-frac(c*phi)) = frac(e*phi); e = d+b;

(IIb) (1-frac(b*phi))*frac(c*phi) = frac(f*phi); f = d+c;

(III) (1-frac(b*phi))*(1-frac(c*phi)) = 1-frac(g*phi); g = d+b+c;

where frac() = FractionalPart(), phi = (1+sqrt(5))/2 and b,c are positive integers.

The parameters d,e,f,g are also positive integers.

Examples

			For b=3, c=10, a(3)=-1, a(10)=2 are solutions of upper formulas:
(I) frac(3*phi)*frac(10*phi) = 1-frac(58*phi); d = 2*3*10+a(3)*10/2+a(10)*3/2 = 58;
(IIa)  frac(3*phi)*(1-frac(10*phi)) = frac(61*phi); e = d+3 = 61;
(IIb)  (1-frac(3*phi))*frac(10*phi) = frac(68*phi); f = d+10 = 68;
(III)  (1-frac(3*phi))*(1-frac(10*phi)) = 1-frac(71*phi); g = d+3+10 = 71.
		

Crossrefs

Cf. A000201, A001622, A005206, A050140, A189663, A019446, A060144.

Programs

Mathematica

Table[2Floor[n*GoldenRatio]-3n,{n,76}] (* Stefano Spezia, Dec 18 2020 *)

PARI

a(n) = 2*floor(n*quadgen(5)) - 3*n; \\ Michel Marcus, Jan 05 2021

Python

from math import isqrt
def A339765(n): return ((n+isqrt(5*n**2))&~1)-3*n # Chai Wah Wu, Aug 09 2022

Formula

a(n) = 2*A000201(n) - 3*n.

a(n) = A005206(n-1) - A189663(n+1).

a(n) = A019446(n) - A060144(n) - sign(abs(n)) - 1.

From Primoz Pirnat, May 15 2024: (Start)

a(n) = A050140(n) - 2*n.

a(n) = 2*A005206(n-1) - n.

a(n) = n - 2*A189663(n+1). (End)