A028884 - OEIS

%I #78 Feb 05 2024 02:24:52

%S 1,8,17,28,41,56,73,92,113,136,161,188,217,248,281,316,353,392,433,

%T 476,521,568,617,668,721,776,833,892,953,1016,1081,1148,1217,1288,

%U 1361,1436,1513,1592,1673,1756,1841,1928,2017,2108,2201,2296,2393

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

%C From _Klaus Purath_, Jan 04 2023: (Start)

%C 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.

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

%C 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).

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

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

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

%C (End)

%H Altug Alkan, <a href="/A028884/b028884.txt">Table of n, a(n) for n = 0..10000</a>

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + 2*n + 5 (with a(0) = 1). - _Vincenzo Librandi_, Aug 05 2010

%F a(n) = A028560(n) + 1; A014616(n) = floor(a(n+1)/4). - _Reinhard Zumkeller_, Apr 07 2013

%F G.f.: (-1 - 5*x + 4*x^2)/(x - 1)^3. - _R. J. Mathar_, Mar 24 2013

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

%F E.g.f.: (1 + 7*x + x^2)*exp(x). - _G. C. Greubel_, Aug 19 2017

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

%F From _Klaus Purath_, Jan 04 2023: (Start)

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

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

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

%F From _Amiram Eldar_, Feb 05 2024: (Start)

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

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

%e From _Stefano Spezia_, Nov 08 2022: (Start)

%e Illustrations for n = 0..4:

%e           *       * * *     * * * * *

%e       a(0) = 1    *   *     *       *

%e                   * * *     *   *   *

%e                 a(1) = 8    *       *

%e                             * * * * *

%e                             a(2) = 17

%e .

%e    * * * * * * *    * * * * * * * * *

%e    *           *    *               *

%e    *   *   *   *    *   *   *   *   *

%e    *           *    *               *

%e    *   *   *   *    *   *   *   *   *

%e    *           *    *               *

%e    * * * * * * *    *   *   *   *   *

%e      a(3) = 28      *               *

%e                     * * * * * * * * *

%e                         a(4) = 41

%e (End)

%t Range[3, 50]^2 - 8 (* _Alonso del Arte_, Aug 15 2016 *)

%o (Haskell) a014616 n = (n * (n + 6) + 1) `div` 4 -- _Reinhard Zumkeller_, Apr 07 2013

%o (PARI) a(n)=(n+3)^2-8 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Scala) (3 to 49).map(n => n * n - 8) // _Alonso del Arte_, May 07 2020

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

%K nonn,easy

%O 0,2

%A _Patrick De Geest_

%E Definition corrected by _Omar E. Pol_, Jul 27 2009