A133265 Diagonal of the A135356 triangle.

2, 2, 2, 4, 2, 6, 2, 8, 2, 10, 2, 12, 2, 14, 2, 16, 2, 18, 2, 20, 2, 22, 2, 24, 2, 26, 2, 28, 2, 30, 2, 32, 2, 34, 2, 36, 2, 38, 2, 40, 2, 42, 2, 44, 2, 46, 2, 48, 2, 50, 2, 52, 2, 54, 2, 56, 2, 58, 2, 60, 2, 62, 2, 64, 2, 66, 2, 68, 2, 70, 2, 72, 2, 74, 2, 76, 2, 78, 2, 80

Offset: 0

Author

Paul Curtz, Dec 20 2007

Keywords

• nonn
• easy

Comments

Regular continued fraction expansion of 2*sin(1/2)/( cos(1/2) - sin(1/2) ) = 2.40822 34423 35827 84841 ... = 2 + 1/(2 + 1/(2 + 1/(4 + 1/(2 + 1/(6 + 1/(2 + 1/(8 + 1/(2 + ... )))))))). Cf. A019425. - Peter Bala, Feb 15 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A019425.

Programs

Magma

[(n+3+(n-1)*(-1)^(n+1))/2: n in [0..80]]; // Vincenzo Librandi, Aug 30 2011

Maple

A133265 := n -> (n+2+(n-2)*(-1)^n)/2: # Peter Luschny, Aug 30 2011

Mathematica

Table[(n + 3 + (n - 1) (-1)^(n + 1))/2, {n, 0, 79}] (* or *)
Table[Mod[(2 n + 5), (n (3 + (-1)^n) - (-1)^n + 7)/2], {n, 0, 79}] (* or *)
CoefficientList[Series[2 (1 + x - x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 79}], x] (* Michael De Vlieger, Nov 18 2016 *)

PARI

Vec(2*(1 + x - x^2) / ((1 - x)^2 * (1 + x)^2) + O(x^100)) \\ Colin Barker, Nov 17 2016

Formula

2*(A057979 without 1, 0, first two terms).

a(n) = (n+3+(n-1)*(-1)^(n+1))/2. - Vincenzo Librandi, Aug 30 2011

a(n) = (2*n + 5) mod (n*(3 + (-1)^n) - (-1)^n + 7)/2. - Lechoslaw Ratajczak, Nov 17 2016

From Colin Barker, Nov 17 2016: (Start)

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

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

(End)