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
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).
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 *)
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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
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
Comments