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A165719
Integers of the form k*(k+9)/8.
2
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
1,1
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.
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Definition simplified by R. J. Mathar, Sep 25 2009
STATUS
approved