A094024 Alternating 1 with one less than the powers of 2.

1, 1, 1, 3, 1, 7, 1, 15, 1, 31, 1, 63, 1, 127, 1, 255, 1, 511, 1, 1023, 1, 2047, 1, 4095, 1, 8191, 1, 16383, 1, 32767, 1, 65535, 1, 131071, 1, 262143, 1, 524287, 1, 1048575, 1, 2097151, 1, 4194303, 1, 8388607, 1, 16777215, 1, 33554431, 1, 67108863, 1

Offset: 0

Author

Paul Barry, Apr 22 2004

Keywords

• easy
• nonn

Comments

Inverse binomial transform of A052542. Partial sums are A075427.

Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/2) = 0.64227 25013 85234 96714 ... = 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(7 + 1/(1 + 1/(15 + ...)))))))). See A111317. - Peter Bala, Dec 26 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Crossrefs

Cf. A000129, A016116.

Cf. A111317.

Programs

Magma

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

Mathematica

LinearRecurrence[{-1, 2, 2}, {1, 1, 1}, 60] (* Jean-François Alcover, Jul 02 2018 *)

PARI

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

Formula

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

E.g.f.: exp(-x) + 2*sinh(sqrt(2)*x) / sqrt(2).

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

a(n) = (1-(-1)^n) * 2^floor(n/2) + (-1)^n. - Ralf Stephan, Aug 19 2013

a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3). - Andrew Howroyd, Feb 21 2018

Extensions

Better name from Ralf Stephan, Aug 19 2013

Even terms for n >= 60 corrected in b-file by Andrew Howroyd, Feb 21 2018