cp's OEIS Frontend

A218987 Power ceiling sequence of 2+sqrt(7).

%I A218987 #14 Aug 05 2025 16:19:54
%S A218987 5,24,112,521,2421,11248,52256,242769,1127845,5239688,24342288,
%T A218987 113088217,525379733,2440783584,11339273536,52679444897,244735600197,
%U A218987 1136980735480,5282129742512,24539461176489,114004233933493,529635319263440,2460553978854240
%N A218987 Power ceiling sequence of 2+sqrt(7).
%C A218987 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 = 2+sqrt(7), and the limit p4(r) = 5.19758760498048832156707270895307875397561324042...
%C A218987 See A218986 for the power floor function, p1(x); for comparison of p1 and p4, limit(p4(r)/p1(r) = 4 - sqrt(7).
%H A218987 Clark Kimberling, <a href="/A218987/b218987.txt">Table of n, a(n) for n = 0..250</a>
%H A218987 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1,-3).
%F A218987 a(n) = ceiling(x*a(n-1)), where x=2+sqrt(7), a(0) = ceiling(x).
%F A218987 a(n) = 5*a(n-1) - a(n-2) - 3*a(n-3).
%F A218987 G.f.: (5 - x - 3*x^2)/(1 - 5*x + x^2 + 3*x^3).
%F A218987 a(n) = (-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84. - _Colin Barker_, Sep 02 2016
%F A218987 E.g.f.: exp(x)*(exp(x)*(217*cosh(sqrt(7)*x) + 83*sqrt(7)*sinh(sqrt(7)*x)) - 7)/42. - _Stefano Spezia_, Aug 05 2025
%e A218987 a(0) = ceiling(r) = 5, where r = 2+sqrt(7);
%e A218987 a(1) = ceiling(5*r) = 24; a(2) = ceiling(24*r) = 112.
%t A218987 (See A218986.)
%o A218987 (PARI) a(n) = round((-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84) \\ _Colin Barker_, Sep 02 2016
%o A218987 (PARI) Vec((5-x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ _Colin Barker_, Sep 02 2016
%Y A218987 Cf. A214992, A015530, A126473, A218986.
%K A218987 nonn,easy
%O A218987 0,1
%A A218987 _Clark Kimberling_, Nov 11 2012