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

0,2

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).

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

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 *)

Prog

(Magma) [2^n/3 +2*(-1)^n/3-2: n in [0..40]]; // Vincenzo Librandi, Aug 07 2011
(PARI) a(n)=(2^n+2*(-1)^n-6)/3 \\ Charles R Greathouse IV, Aug 28 2016

Crossrefs

Cf. A000975, A005186, A033491.

Sequence in context: A021837 A236934 A155719 * A197007 A011449 A231037

Adjacent sequences: A153769 A153770 A153771 * A153773 A153774 A153775

Keyword

easy,sign

Author

Paul Curtz, Jan 01 2009

Status

approved