A153772 a(n) = (2^n + 2*(-1)^n - 6)/3.

-1, -2, 0, 0, 4, 8, 20, 40, 84, 168, 340, 680, 1364, 2728, 5460, 10920, 21844, 43688, 87380, 174760, 349524, 699048, 1398100, 2796200, 5592404, 11184808, 22369620, 44739240, 89478484, 178956968, 357913940, 715827880

Offset: 0

Author

Paul Curtz, Jan 01 2009

Keywords

• easy
• sign

Comments

The array of T(n,k) with T(0,k) = A141325(k) and successive differences T(n,k) = T(n-1,k+1) - T(n-1,k) in further rows is

1, 1, 1, 1, 3, 5, 9, 13, 21, 33, 55,..

0, 0, 0, 2, 2, 4, 4, 8, 12, 22,..

0, 0, 2, 0, 2, 0, 4, 4, 10,...

0, 2, -2, 2, -2, 4, 0, 6,..

2, -4, 4, -4, 6, -4, 6,..

-6, 8, -8, 10, -10, 10,...

with T(n,n) = A078008(n), T(n,n+1) = -A167030(n), T(n,n+2) = A128209(n), T(n,n+3) = -a(n). All these sequences along the diagonals obey the recurrences a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) and a(n) = 5*a(n-2) - 4*a(n-4).

Conjecture: For n >= 6, a(n) is the third largest natural number whose Collatz orbit has length n+2. - Markus Sigg, Sep 14 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A000975, A005186, A033491.

Programs

Magma

[2^n/3 +2*(-1)^n/3-2: n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

Mathematica

Table[(2^n + 2*(-1)^n - 6)/3, {n,0,25}] (* or *) LinearRecurrence[{2, 1, -2}, {-1, -2, 0}, 25] (* G. C. Greubel, Aug 27 2016 *)

PARI

a(n)=(2^n+2*(-1)^n-6)/3 \\ Charles R Greathouse IV, Aug 28 2016

Formula

a(n) = A078008(n) - 2.

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

a(n) = a(n-1) + 2*a(n-2) + 4.

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

E.g.f.: (1/3)*(2*exp(-x) - 6*exp(x) + exp(2*x)). - G. C. Greubel, Aug 27 2016

a(n) = 4*A000975(n-3) for n >= 3. - Markus Sigg, Sep 14 2020