OFFSET
0,2
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).
For n > 0, this is a proper subsequence of A079896.
(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).
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)
EXAMPLE
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)
MATHEMATICA
Range[3, 50]^2 - 8 (* Alonso del Arte, Aug 15 2016 *)
PROG
(Haskell) a014616 n = (n * (n + 6) + 1) `div` 4 -- Reinhard Zumkeller, Apr 07 2013
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Definition corrected by Omar E. Pol, Jul 27 2009
STATUS
approved