A092811 Expansion of g.f. (1-4*x)/(1-8*x).

1, 4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936, 576460752303423488, 4611686018427387904

Offset: 0

Author

Paul Barry, Mar 10 2004

Keywords

• easy
• nonn

Comments

4th binomial transform of (1,0,16,0,256,...).

Number of compositions of even natural numbers into n parts <= 7. - Adi Dani, May 28 2011

Examples

			From _Adi Dani_, May 28 2011: (Start)
a(2)=32: there are 32 compositions of even natural numbers into 2 parts <= 7:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(3,7),(7,3),(4,6),(6,4),(5,5);
(5,7),(7,5),(6,6);
(7,7).  (End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (8).

Crossrefs

Cf. A001045, A013731 (same sequence omitting initial 1), A055372, A134309.

Programs

Magma

[8^n/2+0^n/2: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

Mathematica

Table[EulerPhi[8^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

PARI

a(n)=max(1,8^n/2) \\ Charles R Greathouse IV, Apr 09 2012

Python

def A092811(n): return 1<<3*n-1 if n else 1 # Chai Wah Wu, Apr 25 2025

Formula

a(n) = 8^n/2 + 0^n/2.

a(n) = A001045(3n+1) - A001045(3n-1) + 0^n/2.

a(n) = A013731(n-1), n > 0. - R. J. Mathar, Sep 08 2008

a(n) = 4 * 8^(n-1), a(0)=1. - Vincenzo Librandi, Jun 16 2011

a(n) = Sum_{k=0..n} A134309(n,k)*4^k = Sum_{k=0..n} A055372(n,k)*3^k. - Philippe Deléham, Feb 04 2012

E.g.f.: (1 + exp(8*x))/2. - Stefano Spezia, May 29 2024