A195025 a(n) = n*(14*n + 3).

0, 17, 62, 135, 236, 365, 522, 707, 920, 1161, 1430, 1727, 2052, 2405, 2786, 3195, 3632, 4097, 4590, 5111, 5660, 6237, 6842, 7475, 8136, 8825, 9542, 10287, 11060, 11861, 12690, 13547, 14432, 15345, 16286, 17255, 18252, 19277, 20330, 21411, 22520, 23657, 24822, 26015

Offset: 0

Author

Omar E. Pol, Oct 13 2011

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 17, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-axis of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

a(k) is a square for k = (3/56)*((449 + 120*sqrt(14))^n + (449 - 120*sqrt(14))^n - 2). - Bruno Berselli, Oct 18 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A144555, A152760, A185019, A193053, A195019, A195020, A195023, A195024, A195320.

Programs

Magma

[14*n^2 +3*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Mathematica

Table[n(14n+3),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,17,62},50] (* Harvey P. Dale, Jul 17 2023 *)

PARI

a(n)=n*(14*n+3) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 14*n^2 + 3*n.

G.f.: x*(17+11*x)/(1-x)^3. - Bruno Berselli, Oct 18 2011

From Elmo R. Oliveira, Dec 30 2024: (Start)

E.g.f.: exp(x)*x*(17 + 14*x).

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

Extensions

Name suggested by Bruno Berselli, Oct 13 2011