A067239 a(0)=1, a(n) = 8*n*(2*n-1).

1, 8, 48, 120, 224, 360, 528, 728, 960, 1224, 1520, 1848, 2208, 2600, 3024, 3480, 3968, 4488, 5040, 5624, 6240, 6888, 7568, 8280, 9024, 9800, 10608, 11448, 12320, 13224, 14160, 15128, 16128, 17160, 18224, 19320, 20448, 21608, 22800, 24024

Offset: 0

Author

Benoit Cloitre, Mar 03 2002

Keywords

• easy
• nonn

Comments

Engel expansion of cosh(1/2).

Also, 8 times hexagonal numbers (8*A000384(n) = A152750(n)), for n > 0. - Omar E. Pol, Dec 14 2008

Links

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

Crossrefs

Cf. A006784 for Engel expansion definition.

Cf. A000384, A152750.

Programs

Mathematica

s=0;lst={s};Do[s+=n++ +8;AppendTo[lst, s], {n, 0, 8!, 32}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

PARI

Vec((1+5*x+27*x^2-x^3)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Apr 13 2012

Formula

a(n) = a(n-1) + 32*n - 24 (with a(1)=8). - Vincenzo Librandi, Dec 15 2010

From Colin Barker, Apr 13 2012: (Start)

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

G.f.: (1 + 5*x + 27*x^2 - x^3)/(1-x)^3. (End)

E.g.f.: 1 + 8*exp(x)*x*(1 + 2*x). - Elmo R. Oliveira, Dec 15 2024

From Amiram Eldar, May 05 2025: (Start)

Sum_{n>=0} 1/a(n) = log(2)/4 + 1.

Sum_{n>=0} (-1)^n/a(n) = 1 - Pi/16 + log(2)/8. (End)