A214998 Power ceiling-floor sequence of 2 + sqrt(3).

4, 14, 53, 197, 736, 2746, 10249, 38249, 142748, 532742, 1988221, 7420141, 27692344, 103349234, 385704593, 1439469137, 5372171956, 20049218686, 74824702789, 279249592469, 1042173667088, 3889445075882, 14515606636441, 54172981469881, 202176319243084

Offset

0,1

Comments

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2 + sqrt(3), and the limit p3(r) = (23 + 13*sqrt(3))/12.

References

R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.

Links

Clark Kimberling, Table of n, a(n) for n = 0..250

Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Formula

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1) if n is even, where x = 2+sqrt(3) and a(0) = ceiling(x).

a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3).

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

a(n) = (-1)^n + 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 14. - Peter Bala, Nov 12 2017

a(n) = (1/12)*(2*(-1)^n + (23-13*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(23+13*sqrt(3))). - Colin Barker, Nov 13 2017

Example

a(0) = ceiling(r) =  4, where r = 2+sqrt(3);
a(1) = floor(4*r) = 14; a(2) = ceiling(14*r) = 53.

Mathematica

x = 2 + Sqrt[3]; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A001835 *)
Table[p2[n], {n, 0, z}]  (* A109437 *)
Table[p3[n], {n, 0, z}]  (* A214998 *)
Table[p4[n], {n, 0, z}]  (* A001353 *)

Prog

(PARI) Vec((4 + 2*x - x^2) / ((1 + x)*(1 - 4*x + x^2)) + O(x^30)) \\ Colin Barker, Nov 13 2017

Crossrefs

Cf. A214992, A001835, A109437, A001353, A001654.

Sequence in context: A295518 A047118 A017948 * A112872 A162482 A308555

Adjacent sequences: A214995 A214996 A214997 * A214999 A215000 A215001

Keyword

nonn,easy

Author

Clark Kimberling, Nov 10 2012

Status

approved