A182615 Greatest k such that floor(k/r^n)=n, where r = golden mean = (1+sqrt(5))/2.

3, 7, 16, 34, 66, 125, 232, 422, 760, 1352, 2388, 4185, 7294, 12644, 21824, 37518, 64278, 109781, 186980, 317666, 538472, 910868, 1537896, 2592049, 4361786, 7328960, 12297712, 20608762, 34495530, 57675437, 96331168, 160737950, 267960664, 446321504, 742796604, 1235255433

Offset

1,1

Links

Table of n, a(n) for n=1..36.

Formula

For n>=3, a(n)=-1+A182614(n)+A000032(n), where A000032 is the sequence of Lucas numbers.

Conjectures from Chai Wah Wu, Jan 12 2023: (Start)

a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) - 2*a(n-4) + 2*a(n-5) + a(n-6) for n > 8.

G.f.: x*(-x^7 - x^6 + 3*x^5 + 4*x^2 - x - 3)/((x - 1)*(x + 1)*(x^2 + x - 1)^2). (End)

Example

The integers k satisfying floor(k/r^3)=3 are 13,14,15,16, so that a(3)=16.

Prog

(PARI) a(n) = floor(((1+sqrt(5))/2)^n*(n+1)) \\ David A. Corneth, May 07 2022

Crossrefs

Cf. A001622, A182614, A000032.

Sequence in context: A002936 A014668 A354909 * A181893 A054455 A178455

Adjacent sequences: A182612 A182613 A182614 * A182616 A182617 A182618

Keyword

nonn

Author

Clark Kimberling, Nov 22 2010

Extensions

a(23) corrected by Andrey Zabolotskiy, May 07 2022

More terms from David A. Corneth, May 07 2022

Status

approved