A112087 a(n) = 4*(n^2 - n + 1).

4, 12, 28, 52, 84, 124, 172, 228, 292, 364, 444, 532, 628, 732, 844, 964, 1092, 1228, 1372, 1524, 1684, 1852, 2028, 2212, 2404, 2604, 2812, 3028, 3252, 3484, 3724, 3972, 4228, 4492, 4764, 5044, 5332, 5628, 5932, 6244, 6564, 6892, 7228, 7572, 7924, 8284

Offset: 1

Author

Jenny X. Chen (pookahead(AT)gmail.com), Nov 28 2005

Keywords

• nonn
• easy

Comments

Diagonal of a number spiral.

Examples

			..43.44.45.46.47.48.49.50...
..42.21.22.23.24.25.26.51...
..40.19..7..8..9.10.27.52...
..40.19..6..1..2.11.28....
..39.18..5..4..3.12.29
..38.17.16.15.14.13.30
..37.36.35.34.33.32.31
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002061.

Programs

Mathematica

Table[4(n^2 - n + 1), {n, 2, 47}] (* Robert G. Wilson v, Nov 29 2005 *)

Maxima

makelist(4*n^2 - 4*n + 4,n,1,55); /* Martin Ettl, Oct 16 2012 */

PARI

a(n)=4*(n^2-n+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 4*A002061(n). - Robert G. Wilson v, Nov 29 2005

a(n) = 8*n + a(n-1) - 8 (with a(1)=4). - Vincenzo Librandi, Nov 13 2010

Sum_{n>=1} 1/a(n) = tanh(Pi*sqrt(3)/2)*Pi/(4*sqrt(3)). - Amiram Eldar, Aug 25 2022

From Elmo R. Oliveira, Feb 08 2025: (Start)

G.f.: 4*x*(1 + x^2)/(1 - x)^3.

E.g.f.: -4 + 4*exp(x)*(1 + x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)

Extensions

Edited and extended by Robert G. Wilson v, Nov 29 2005