A166519 a(n) = 1 + 2*(-1)^n + 2*n.

3, 1, 7, 5, 11, 9, 15, 13, 19, 17, 23, 21, 27, 25, 31, 29, 35, 33, 39, 37, 43, 41, 47, 45, 51, 49, 55, 53, 59, 57, 63, 61, 67, 65, 71, 69, 75, 73, 79, 77, 83, 81, 87, 85, 91, 89, 95, 93, 99, 97, 103, 101, 107, 105, 111, 109, 115, 113, 119, 117, 123, 121, 127, 125, 131, 129, 135

Offset: 0

Author

Vincenzo Librandi, Oct 16 2009

Keywords

• nonn
• easy

Comments

Many pairs of primes of the form p+6 (5,11 - 13,19 - 17,23 - 37,43 - 41,47 - 53,59 - 61,67 - 73,79 - 97,103 - 101,107 - and so on).

a(n) is A005408(n) = 1+2*n swapped by pairs. - Paul Curtz, Mar 07 2011

Links

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

Crossrefs

Cf. A005408.

Programs

Magma

[1 + 2*(-1)^n + 2*n: n in [0..70]]; // Vincenzo Librandi, Dec 01 2012

Mathematica

Table[1 + 2*(-1)^n + 2*n, {n, 0, 100}] (* Vincenzo Librandi, Dec 01 2012 *)
LinearRecurrence[{1,1,-1},{3,1,7},70] (* Harvey P. Dale, Jan 16 2023 *)

PARI

vector(100, n, n--; 1 + 2*(-1)^n + 2*n) \\ Altug Alkan, Oct 19 2015

SageMath

def A166519(n): return 4*((n+1)%2) + 2*n -1
[A166519(n) for n in range(101)] # G. C. Greubel, Aug 03 2024

Formula

a(n) = 4*n - a(n-1), n >= 1.

G.f.: ( 3-2*x+3*x^2 ) / ( (1+x)*(1-x)^2 ). - R. J. Mathar, Nov 02 2011

a(n) = a(n-1) + a(n-2) - a(n-3). - Vincenzo Librandi, Dec 01 2012

E.g.f.: (1 + 2*x)*exp(x) + 2*exp(-x). - G. C. Greubel, May 16 2016

Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/4. - Amiram Eldar, Mar 02 2023