login
A165717
Integers of the form k*(5+k)/4.
6
6, 9, 21, 26, 44, 51, 75, 84, 114, 125, 161, 174, 216, 231, 279, 296, 350, 369, 429, 450, 516, 539, 611, 636, 714, 741, 825, 854, 944, 975, 1071, 1104, 1206, 1241, 1349, 1386, 1500, 1539, 1659, 1700, 1826, 1869, 2001, 2046, 2184, 2231, 2375, 2424, 2574, 2625
OFFSET
1,1
COMMENTS
Integers of the form k+k*(k+1)/4 = k+A000217(k)/2; for k see A014601, for A000217(k)/2 see A074378.
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
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
EXTENSIONS
Definition simplified by R. J. Mathar, Sep 25 2009
STATUS
approved