A259821 a(n) = floor( (3^n+1)^2/3^n ).

4, 5, 11, 29, 83, 245, 731, 2189, 6563, 19685, 59051, 177149, 531443, 1594325, 4782971, 14348909, 43046723, 129140165, 387420491, 1162261469, 3486784403, 10460353205, 31381059611, 94143178829, 282429536483

Offset: 0

Author

Kival Ngaokrajang, Jul 07 2015

Keywords

• nonn
• easy
• less

Comments

a(n) is the curvature (rounded down) of circles inscribed between a unit circle and a circumscribed equilateral triangle. See illustration.

Apart from the first term the same as A168607. - R. J. Mathar, Jul 09 2015

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A034472, A000244, A168607.

Programs

Mathematica

Table[Floor[(3^n + 1)^2/3^n], {n, 0, 30}] (* Michael De Vlieger, Jul 07 2015 *)
LinearRecurrence[{4,-3},{4,5,11},30] (* Harvey P. Dale, Sep 30 2024 *)

PARI

a(n)=floor((3^n+1)^2/3^n)
for (n=0, 100, print1(a(n),", "))

PARI

Vec((3*x^2-11*x+4)/((x-1)*(3*x-1)) + O(x^100)) \\ Colin Barker, Jul 07 2015

Formula

a(n) = floor( A034472(n)^2/A000244(n) ).

From Colin Barker, Jul 07 2015: (Start)

a(n) = 3^n + 2 for n>0.

a(n) = 4*a(n-1) - 3*a(n-2) for n>2.

G.f.: (3*x^2-11*x+4) / ((x-1)*(3*x-1)).

(End)