A165721 Integers of the form k*(k+13)/12.

4, 14, 22, 25, 35, 55, 69, 74, 90, 120, 140, 147, 169, 209, 235, 244, 272, 322, 354, 365, 399, 459, 497, 510, 550, 620, 664, 679, 725, 805, 855, 872, 924, 1014, 1070, 1089, 1147, 1247, 1309, 1330, 1394, 1504, 1572, 1595, 1665, 1785, 1859, 1884, 1960, 2090

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

Keywords

• nonn
• easy

Comments

Integers of the form k+k*(k+1)/12 = k+A000217(k)/6 (see A069497). - R. J. Mathar, Sep 25 2009

Are all terms composite numbers?

Contribution from Zak Seidov, Sep 25 2009: (Start)

Integers of form n(13+n)/12, n=0,1,2,...

Each four terms of the sequence are composite numbers of forms:

{(4+3 m) (1+4 m), (2+3 m) (7+4 m), (2+m) (11+12 m), m (13+12 m)}, m=0,1,2,...

m=0: {4,14,22,25}; m=1: {35,55,69,74}; m=2: {90,120,140,147}, etc. (End)

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Crossrefs

Cf. A000217, A069497, A165717, A165718, A165719, A165720.

Programs

Mathematica

q=6;s=0;lst={};Do[s+=((n+q)/q);If[IntegerQ[s],AppendTo[lst,s]],{n,6!}];lst
Select[Table[k (k+13)/12,{k,200}],IntegerQ] (* or *) LinearRecurrence[ {3,-5,7,-7,5,-3,1},{4,14,22,25,35,55,69},50] (* Harvey P. Dale, Jan 30 2013 *)

Formula

From R. J. Mathar, Sep 25 2009: (Start)

a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7).

G.f.: x*(-4-2*x-x^3+x^5)/((x^2+1)^2*(x-1)^3). (End)

Sum_{n>=1} 1/a(n) = 712/507 - (3 + 4*sqrt(3))*Pi/39. - Amiram Eldar, Jul 26 2024

Extensions

Definition simplified by R. J. Mathar, Sep 25 2009