A344275 Denominators of worst rational approximations to the golden ratio.

1, 4, 9, 12, 17, 22, 25, 30, 38, 43, 51, 59, 64, 72, 80, 85, 93, 106, 114, 127, 140, 148, 161, 174, 182, 195, 216, 229, 237, 250, 271, 284, 305, 326, 339, 360, 373, 394, 415, 428, 449, 483, 504, 538, 559, 572, 593, 627, 648, 682, 716, 737, 771, 792, 826, 860, 881

Offset

1,2

Comments

These are positive integers k such that ||k*phi||/k > ||m*phi||/m for all integers m > k, where phi = (1+sqrt(5))/2 is the golden ratio, and ||x|| is the distance from x to the nearest integer. Luke Schaeffer (personal communication) asks for a formula for these.

Conjecture: difference of consecutive terms is always a Fibonacci number.

Links

Table of n, a(n) for n=1..57.

Crossrefs

Cf. A000045, A001622, A344290 (numerators).

Sequence in context: A301688 A276885 A089910 * A312862 A177880 A059269

Adjacent sequences: A344272 A344273 A344274 * A344276 A344277 A344278

Keyword

nonn,frac

Author

Jeffrey Shallit, May 13 2021

Status

approved