A028884 a(n) = (n + 3)^2 - 8.

1, 8, 17, 28, 41, 56, 73, 92, 113, 136, 161, 188, 217, 248, 281, 316, 353, 392, 433, 476, 521, 568, 617, 668, 721, 776, 833, 892, 953, 1016, 1081, 1148, 1217, 1288, 1361, 1436, 1513, 1592, 1673, 1756, 1841, 1928, 2017, 2108, 2201, 2296, 2393

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

From Klaus Purath, Jan 04 2023: (Start)

The product of two consecutive terms belongs to the sequence: a(n)*a(n+1) = a(a(n)+n) = (a(n)+n)*(a(n+1)-n-1) + 1.

a(n) is never divisible by primes given in A003629.

Each odd prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -6 (mod p).

The prime factors are listed in A038873 and the primes in A028886.

For n > 0, this is a proper subsequence of A079896.

Conjecture: a(n) = A079896(A265284(n-1)). -

(End)

Examples

			From _Stefano Spezia_, Nov 08 2022: (Start)
Illustrations for n = 0..4:
          *       * * *     * * * * *
      a(0) = 1    *   *     *       *
                  * * *     *   *   *
                a(1) = 8    *       *
                            * * * * *
                            a(2) = 17
.
   * * * * * * *    * * * * * * * * *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   * * * * * * *    *   *   *   *   *
     a(3) = 28      *               *
                    * * * * * * * * *
                        a(4) = 41
(End)
		

Links

• Altug Alkan, Table of n, a(n) for n = 0..10000
• Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005563, A014616, A028560, A082111, A190576.

Programs

Haskell

a014616 n = (n * (n + 6) + 1) `div` 4 -- Reinhard Zumkeller, Apr 07 2013

Mathematica

Range[3, 50]^2 - 8 (* Alonso del Arte, Aug 15 2016 *)

PARI

a(n)=(n+3)^2-8 \\ Charles R Greathouse IV, Oct 07 2015

Scala

(3 to 49).map(n => n * n - 8) // Alonso del Arte, May 07 2020

Formula

a(n) = a(n-1) + 2*n + 5 (with a(0) = 1). - Vincenzo Librandi, Aug 05 2010

a(n) = A028560(n) + 1; A014616(n) = floor(a(n+1)/4). - Reinhard Zumkeller, Apr 07 2013

G.f.: (-1 - 5*x + 4*x^2)/(x - 1)^3. - R. J. Mathar, Mar 24 2013

Sum_{n >= 0} 1/a(n) = 51/112 - Pi*cot(2*Pi*sqrt(2))/(4*sqrt(2)) = 1.3839174974448... . - Vaclav Kotesovec, Apr 10 2016

E.g.f.: (1 + 7*x + x^2)*exp(x). - G. C. Greubel, Aug 19 2017

Sum_{n >= 0} (-1)^n/a(n) = (-19 + 14*sqrt(2)*Pi*cosec(2*sqrt(2)*Pi))/112. - Amiram Eldar, Nov 04 2020

From Klaus Purath, Jan 04 2023: (Start)

a(n) = 2*a(n-1) - a(n-2) + 2, n >= 2.

a(n) = A082111(n) + n.

a(n) = A190576(n+1) - n. (End)

From Amiram Eldar, Feb 05 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = 7*Pi/(45*sqrt(2)*sin(2*sqrt(2)*Pi)).

Product_{n>=0} (1 + 1/a(n)) = (4*sqrt(14)/9)*sin(sqrt(7)*Pi)/sin(2*sqrt(2)*Pi). (End)

Extensions

Definition corrected by Omar E. Pol, Jul 27 2009