A165719 -id:A165719 - cp's OEIS frontend

A165720 Integers of the form k*(k+11)/10.

6, 8, 18, 21, 35, 39, 57, 62, 84, 90, 116, 123, 153, 161, 195, 204, 242, 252, 294, 305, 351, 363, 413, 426, 480, 494, 552, 567, 629, 645, 711, 728, 798, 816, 890, 909, 987, 1007, 1089, 1110, 1196, 1218, 1308, 1331, 1425, 1449, 1547, 1572, 1674, 1700, 1806

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

Integers of the form k + k*(k+1)/10 = k + A000217(k)/5. For k see A047208, for A000217(k)/5 see A057569. - R. J. Mathar, Sep 25 2009
Are all terms composite numbers?
Yes. They are alternately of the form (h+2)*(5*h-1)/2 and h*(5*h+11)/2, with h>0. - Bruno Berselli, Dec 22 2016

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A047208, A057569, A165717, A165718, A165719, A279895.

Select[k = Range[0, 130]; k (k + 11)/10, IntegerQ] (* Bruno Berselli, Dec 22 2016 *)

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).
a(n) = 5*(2*n^2 + 10*n + 3)/16 - 3*(-1)^n*(5 + 2*n)/16.
G.f.: x*(-6 - 2*x + 2*x^2 + x^3) / ((1 + x)^2*(x - 1)^3). (End)
Sum_{n>=1} 1/a(n) = 514/363 - 2*Pi*sqrt(1+2/sqrt(5))/11. - Amiram Eldar, Jul 26 2024

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

Corrected A-number in my comment - R. J. Mathar, Oct 30 2009

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

Original entry on oeis.org

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

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

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)

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

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

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 *)

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

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

cp's OEIS Frontend

A165720 Integers of the form k*(k+11)/10.

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