A024037 a(n) = 4^n - n.

1, 3, 14, 61, 252, 1019, 4090, 16377, 65528, 262135, 1048566, 4194293, 16777204, 67108851, 268435442, 1073741809, 4294967280, 17179869167, 68719476718, 274877906925, 1099511627756, 4398046511083, 17592186044394, 70368744177641, 281474976710632, 1125899906842599

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), this sequence (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Cf. A140660 (first differences).

Programs

Magma

[4^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

Magma

I:=[1, 3, 14]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

Mathematica

Table[4^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 5 x^2) / ((1 - 4 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)

PARI

a(n)=4^n-n \\ Charles R Greathouse IV, Sep 24 2015

Formula

From Vincenzo Librandi, Jun 16 2013: (Start)

G.f.: (1 - 3*x + 5*x^2)/((1 - 4*x)*(1 - x)^2).

a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). (End)

E.g.f.: exp(x)*(exp(3*x) - x). - Elmo R. Oliveira, Sep 10 2024