A195024 a(n) = n*(14*n - 1).

0, 13, 54, 123, 220, 345, 498, 679, 888, 1125, 1390, 1683, 2004, 2353, 2730, 3135, 3568, 4029, 4518, 5035, 5580, 6153, 6754, 7383, 8040, 8725, 9438, 10179, 10948, 11745, 12570, 13423, 14304, 15213, 16150, 17115, 18108, 19129, 20178, 21255, 22360, 23493, 24654, 25843

Offset: 0

Author

Omar E. Pol, Oct 13 2011

Keywords

• nonn
• easy

Comments

Related to the primitive Pythagorean triple [3, 4, 5].

Sequence found by reading the line from 0, in the direction 0, 13, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral.

Also sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - Omar E. Pol, Jul 28 2012

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. A118277, A144555, A152760, A185019, A193053, A195019, A195020, A195320, A198017.

Programs

Magma

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

Mathematica

Table[n(14n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,54},50] (* Harvey P. Dale, Jul 28 2012 *)

PARI

a(n)=n*(14*n-1) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 14*n^2 - n.

From Colin Barker, Apr 09 2012: (Start)

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

G.f.: x*(13+15*x)/(1-x)^3. (End)

E.g.f.: exp(x)*x*(13 + 14*x). - Elmo R. Oliveira, Jan 12 2025