A112032 Denominator of 3/4 + 1/4 - 3/8 - 1/8 + 3/16 + 1/16 - 3/32 - 1/32 + 3/64 ...

4, 1, 8, 2, 16, 4, 32, 8, 64, 16, 128, 32, 256, 64, 512, 128, 1024, 256, 2048, 512, 4096, 1024, 8192, 2048, 16384, 4096, 32768, 8192, 65536, 16384, 131072, 32768, 262144, 65536, 524288, 131072, 1048576, 262144, 2097152, 524288, 4194304, 1048576

Offset: 0

Author

Reinhard Zumkeller, Aug 27 2005

Keywords

• nonn
• frac

Comments

Denominator of partial sums of A112030(n)/A016116(n+4), numerators = A112031;

A112031(n)/a(n) - 2/3 = (-1)^floor(n/2) / A112033(n);

lim_{n->infinity} A112031(n)/a(n) = 2/3.

References

• G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 4, Sect. 1, Problem 148.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (0,2).

Crossrefs

Cf. A016116, A112030, A112031, A112033.

Programs

Magma

[2^(Floor(n/2) + 1 + (-1)^n): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

Mathematica

LinearRecurrence[{0,2},{4,1},50] (* following conjecture in Formula field above *) (* Harvey P. Dale, Dec 21 2014 *)

PARI

m=50; v=concat([4,1], vector(m-2)); for(n=3, m, v[n]=2*v[n-2]); v \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = 2^(floor(n/2) + 1 + (-1)^n) = 2^A084964(n).

Conjectures from Colin Barker, Apr 05 2013: (Start)

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

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

Extensions

a(21) corrected by Vincenzo Librandi, Aug 17 2011