A167936 a(n) = 2^n - A108411(n).

0, 1, 1, 5, 7, 23, 37, 101, 175, 431, 781, 1805, 3367, 7463, 14197, 30581, 58975, 124511, 242461, 504605, 989527, 2038103, 4017157, 8211461, 16245775, 33022991, 65514541, 132623405, 263652487, 532087943, 1059392917, 2133134741, 4251920575, 8546887871

Offset: 0

Author

Paul Curtz, Nov 15 2009

Keywords

• nonn
• easy

Comments

The binomial transform of (0 followed by A077917).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Crossrefs

Cf. A005061, A077917, A083658, A085350, A108411, A167762, A167784.

Programs

Magma

I:=[0,1,1]; [n le 3 select I[n] else 2*Self(n-1) +3*Self(n-2) -6*Self(n-3): n in [1..40]]; // G. C. Greubel, Sep 10 2023

Mathematica

LinearRecurrence[{2,3,-6}, {0,1,1}, 50] (* G. C. Greubel, Jul 01 2016 *)

Python

def A167936(n): return (1<>1) # Chai Wah Wu, Nov 14 2023

SageMath

def A167936(n): return 2^n - ((n+1)%2)*3^(n//2) - (n%2)*3^((n-1)//2)
[A167936(n) for n in range(41)] # G. C. Greubel, Sep 10 2023

Formula

a(n) = A167762(n+1) - A167762(n).

a(n+1) - a(n) = A167784(n).

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

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

a(2n) = A005061(n), a(2n+1) = A085350(n).

a(n) - 2*a(n-1) = (-1)^(n+1)*A083658(n+1).

From G. C. Greubel, Sep 10 2023: (Start)

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

E.g.f.: exp(2*x) - cosh(sqrt(3)*x) - (1/sqrt(3))*sinh(sqrt(3)*x). (End)

Extensions

Edited and extended by R. J. Mathar, Feb 27 2010