A165719 Integers of the form k*(k+9)/8.

14, 17, 45, 50, 92, 99, 155, 164, 234, 245, 329, 342, 440, 455, 567, 584, 710, 729, 869, 890, 1044, 1067, 1235, 1260, 1442, 1469, 1665, 1694, 1904, 1935, 2159, 2192, 2430, 2465, 2717, 2754, 3020, 3059, 3339, 3380, 3674, 3717, 4025, 4070, 4392, 4439, 4775

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

Keywords

• nonn
• easy

Comments

Only one term is a prime number (17). Are all others composite?

There is no prime other than 17 in the first 1 million terms. - Harvey P. Dale, Jan 07 2020

Integers of the form k+k*(k+1)/8 = k+A000217(k)/4; for k see A047521, for A000217(k)/4 see A154260.

Examples

			for k = 1,2,..., k(k+9)/8 is 5/4, 11/4, 9/2, 13/2, 35/4, 45/4, 14, 17,.. and the integer values out of these become the sequence.
		

Links

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

Crossrefs

Cf. A000217, A047521, A154260, A165717, A165718.

Programs

Mathematica

q=4;s=0;lst={};Do[s+=((n+q)/q);If[IntegerQ[s],AppendTo[lst,s]],{n,6!}];lst
Select[Table[(n(n+9))/8,{n,200}],IntegerQ] (* or *) Rest[Flatten[Table[ {9n+8n^2,14+23n+8n^2},{n,0,30}]]] (* or *) LinearRecurrence[{1,2,-2,-1,1},{14,17,45,50,92},60] (* Harvey P. Dale, Jan 07 2020 *)

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).

a(n) = 2*n^2 + 6*n + 9/4 - 3*(-1)^n*(2*n+3)/4.

G.f.: x*(-14-3*x+x^3)/((1+x)^2 * (x-1)^3 ). (End)

Sum_{n>=1} 1/a(n) = 89/81 - (sqrt(2)+1)*Pi/9. - Amiram Eldar, Jul 26 2024

Extensions

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