A086341 a(n) = 2*2^floor(n/2) - (-1)^n.

1, 3, 3, 5, 7, 9, 15, 17, 31, 33, 63, 65, 127, 129, 255, 257, 511, 513, 1023, 1025, 2047, 2049, 4095, 4097, 8191, 8193, 16383, 16385, 32767, 32769, 65535, 65537, 131071, 131073, 262143, 262145, 524287, 524289, 1048575, 1048577, 2097151, 2097153

Offset: 0

Author

Paul Barry, Jul 16 2003

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Crossrefs

Cf. A016116 (2^floor(n/2)).

Cf. A065442, A248721, A323482.

Programs

Magma

[2*2^Floor(n/2)-(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Mathematica

CoefficientList[Series[(1+2x)^2/((1+x)(1-2x^2)),{x,0,50}],x] (* or *) LinearRecurrence[ {-1,2,2},{1,3,3},50] (* Harvey P. Dale, Mar 10 2013 *)

PARI

vector(40, n, n--; 2^(floor(n/2)+1) - (-1)^n) \\ G. C. Greubel, Nov 08 2018

Formula

E.g.f.: 2*cosh(sqrt(2)*x) + 2*sinh(sqrt(2)*x)/sqrt(2) - sinh(x) + cosh(x).

a(n) = (1 + 1/sqrt(2))*sqrt(2)^n + (1 - 1/sqrt(2))*(-sqrt(2))^n - (-1)^n.

G.f.: (1+2*x)^2/((1+x)*(1-2*x^2)). - Colin Barker, Aug 17 2012

a(n) = a(n-1) + 2*a(n-2) + 2*a(n-3); a(0)=1, a(1)=3, a(2)=3. - Harvey P. Dale, Mar 10 2013

From Amiram Eldar, Sep 14 2022: (Start)

Sum_{n>=0} 1/a(n) = A065442 + A323482 - 1/2.

Sum_{n>=0} (-1)^n/a(n) = 2 * A248721. (End)