A218986 - OEIS

%I #13 Sep 02 2016 06:00:53

%S 4,18,83,385,1788,8306,38587,179265,832820,3869074,17974755,83506241,

%T 387949228,1802315634,8373110219,38899387777,180716881764,

%U 839565690386,3900413406835,18120350698497,84182643014492,391091624153458,1816914425657307,8440932575089601

%N Power floor sequence of 2+sqrt(7).

%C See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n.  The present sequence is a(n,r), where r = 2+sqrt(7), and the limit p1(r) = 3.83798607113023840500712572585708...

%C See A218987 for the power floor function, p4.  For comparison with p1, limit(p4(r)/p1(r) = 4-sqrt(7).

%H Clark Kimberling, <a href="/A218986/b218986.txt">Table of n, a(n) for n = 0..250</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1,-3).

%F a(n) = [x*a(n-1)], where x=2+sqrt(7), a(0) = [x].

%F a(n) = 5*a(n-1) - a(n-2) - 3*a(n-3).

%F G.f.:  (4 - 2*x - 3*x^2)/(1 - 5*x + x^2 + 3*x^3).

%F a(n) = (14+(161-61*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(161+61*sqrt(7)))/84. - _Colin Barker_, Sep 02 2016

%e a(0) = [r] = 4, where r = 2+sqrt(7);

%e a(1) = [4*r] = 18; a(2) = [18*r] = 83.

%t x = 2 + Sqrt[7]; z = 30; (* z = # terms in sequences *)

%t f[x_] := Floor[x]; c[x_] := Ceiling[x];

%t p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];

%t p1[n_] := f[x*p1[n - 1]]

%t p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]

%t p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]

%t p4[n_] := c[x*p4[n - 1]]

%t t1 = Table[p1[n], {n, 0, z}]  (* A218986 *)

%t t2 = Table[p2[n], {n, 0, z}]  (* A015530 *)

%t t3 = Table[p3[n], {n, 0, z}]  (* A126473 *)

%t t4 = Table[p4[n], {n, 0, z}]  (* A218987 *)

%t LinearRecurrence[{5,-1,-3},{4,18,83},30] (* _Harvey P. Dale_, Jun 18 2014 *)

%o (PARI) a(n) = round((14+(161-61*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(161+61*sqrt(7)))/84) \\ _Colin Barker_, Sep 02 2016

%o (PARI) Vec((4-2*x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ _Colin Barker_, Sep 02 2016

%Y Cf. A214992, A015530, A126473, A218987.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Nov 11 2012