OFFSET
1,1
COMMENTS
Are all terms composite?
Yes, because a(2*k) = k*(4*k+5) and a(2*k-1) = (k+1)*(4*k-1). - Bruno Berselli, Apr 07 2013
Numbers m such that 16*m + 25 is a square. - Vincenzo Librandi, Apr 07 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
From R. J. Mathar, Sep 25 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(-6-3*x+x^3)/( (1+x)^2 * (x-1)^3 ). (End)
Sum_{n>=1} 1/a(n) = 29/25 - Pi/5. - Amiram Eldar, Jul 26 2024
EXAMPLE
For k =1,2,3,.. the value of k*(k+5)/4 is 3/2, 7/2, 6, 9, 25/2, 33/2, 21, 26, 63/2, 75/2, 44, 51,.. and the integer values define the sequence.
MATHEMATICA
q=2; s=0; lst={}; Do[s+=((n+q)/q); If[IntegerQ[s], AppendTo[lst, s]], {n, 6!}]; lst
Select[Table[k*(5+k)/4, {k, 100}], IntegerQ] (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {6, 9, 21, 26, 44}, 60] (* Harvey P. Dale, Aug 11 2011 *)
Select[Range[1, 3000], IntegerQ[Sqrt[16 # + 25]]&] (* Vincenzo Librandi, Apr 07 2013 *)
PROG
(Magma) [n: n in [1..3000] | IsSquare(16*n+25)]; // Vincenzo Librandi, Apr 07 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, Sep 24 2009
EXTENSIONS
Definition simplified by R. J. Mathar, Sep 25 2009
STATUS
approved