cp's OEIS Frontend

A214994 Power ceiling sequence of (golden ratio)^5.

%I A214994 #22 Sep 08 2022 08:46:02
%S A214994 12,134,1487,16492,182900,2028393,22495224,249475858,2766729663,
%T A214994 30683502152,340285253336,3773821288849,41852319430676,
%U A214994 464149335026286,5147495004719823,57086594386944340,633100033261107564,7021186960259127545,77866156596111510560
%N A214994 Power ceiling sequence of (golden ratio)^5.
%C A214994 See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = (golden ratio)^5, and the limit p4(r) = (1/30)*(105+47*sqrt(5)).
%C A214994 See A214993 for the power floor sequence and power floor function, p1. For comparison with p4, we have p4(r)/p1(r) = (5 + 3*sqrt(5))/10.
%H A214994 Clark Kimberling, <a href="/A214994/b214994.txt">Table of n, a(n) for n = 0..250</a>
%H A214994 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-10,-1).
%F A214994 a(n) = ceiling(x*a(n-1)), x=((1+sqrt(5))/2)^5, a(0) = ceiling(x).
%F A214994 a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3).
%F A214994 G.f.: (12 - 10*x - x^2)/(1 - 12*x + 10*x^2 + x^3).
%F A214994 a(n) = (1/550)*(-50 + (3325-1487*sqrt(5))*((11-5*sqrt(5))/2)^n + ((11+5*sqrt(5))/2)^n*(3325+1487*sqrt(5))). - _Colin Barker_, Nov 13 2017
%e A214994 a(0) = ceiling(r) = [11.0902]=12, where r=(1+sqrt(5))^5.
%e A214994 a(1) = ceiling(12) = 134; a(2) = ceiling(134 ) = 1487.
%t A214994 (See A214993.)
%t A214994 LinearRecurrence[{12,-10,-1}, {12,134,1487}, 30] (* _G. C. Greubel_, Feb 01 2018 *)
%o A214994 (PARI) Vec((12 - 10*x - x^2) / ((1 - x)*(1 - 11*x - x^2)) + O(x^40)) \\ _Colin Barker_, Nov 13 2017
%o A214994 (Magma) I:=[12,134,1487]; [n le 3 select I[n] else 12*Self(n-1) - 10*Self(n-2) - Self(n-3): n in [1..30]]; // _G. C. Greubel_, Feb 01 2018
%Y A214994 Cf. A214992, A214993, A049666, A015457.
%K A214994 nonn,easy
%O A214994 0,1
%A A214994 _Clark Kimberling_, Nov 09 2012