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