A182615 - OEIS

%I #24 Jan 12 2023 20:10:49

%S 3,7,16,34,66,125,232,422,760,1352,2388,4185,7294,12644,21824,37518,

%T 64278,109781,186980,317666,538472,910868,1537896,2592049,4361786,

%U 7328960,12297712,20608762,34495530,57675437,96331168,160737950,267960664,446321504,742796604,1235255433

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

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

%F Conjectures from _Chai Wah Wu_, Jan 12 2023: (Start)

%F 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.

%F 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)

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

%o (PARI) a(n) = floor(((1+sqrt(5))/2)^n*(n+1)) \\ _David A. Corneth_, May 07 2022

%Y Cf. A001622, A182614, A000032.

%K nonn

%O 1,1

%A _Clark Kimberling_, Nov 22 2010

%E a(23) corrected by _Andrey Zabolotskiy_, May 07 2022

%E More terms from _David A. Corneth_, May 07 2022