A243138 a(n) = n^2 + 15*n + 13.

13, 29, 47, 67, 89, 113, 139, 167, 197, 229, 263, 299, 337, 377, 419, 463, 509, 557, 607, 659, 713, 769, 827, 887, 949, 1013, 1079, 1147, 1217, 1289, 1363, 1439, 1517, 1597, 1679, 1763, 1849, 1937, 2027, 2119, 2213, 2309, 2407, 2507, 2609, 2713, 2819, 2927, 3037, 3149

Offset: 0

Author

Vincenzo Librandi, Jun 02 2014

Keywords

• nonn
• easy

Comments

From Klaus Purath, Dec 13 2022: (Start)

Numbers m such that 4*m + 173 is a square.

The product of two consecutive terms belongs to the sequence, a(n)*a(n+1) = a(a(n)+n).

The prime terms in this sequence are listed in A153422. Each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -15 (mod p). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A098847, A119412, A126719, A132759, A153422.

Programs

Magma

[n^2+15*n+13: n in [0..50]];

Mathematica

Table[n^2 + 15 n + 13, {n, 0, 50}] (* or *) CoefficientList[Series[(13 - 10 x - x^2)/(1 - x)^3, {x, 0, 50}], x]
LinearRecurrence[{3,-3,1},{13,29,47},50] (* Harvey P. Dale, Sep 06 2020 *)

PARI

a(n)=n^2+15*n+13 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (13 - 10*x - x^2)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.

From Klaus Purath, Dec 13 2022: (Start)

a(n) = A119412(n+2) - 13.

a(n) = A132759(n+1) - 1.

a(n) = A098847(n+1) + n. (End)

Sum_{n>=0} 1/a(n) = tan(sqrt(173)*Pi/2)*Pi/sqrt(173) + 742077303/604626139. - Amiram Eldar, Feb 14 2023

E.g.f.: (13 + 16*x + x^2)*exp(x). - Elmo R. Oliveira, Oct 18 2024