A083313 a(0)=1; a(n) = 3^n - 2^(n-1) for n >= 1.

1, 2, 7, 23, 73, 227, 697, 2123, 6433, 19427, 58537, 176123, 529393, 1590227, 4774777, 14332523, 43013953, 129074627, 387289417, 1161999323, 3486260113, 10459304627, 31378962457, 94138984523, 282421147873, 847271832227, 2541832273897, 7625530376123

Offset: 0

Author

Paul Barry, Apr 24 2003

Keywords

• easy
• nonn

Comments

Essentially the same as A064686.

Binomial transform of A051049.

Number of skinny Boolean functions f(x_1,...,x_n) that are also Horn functions. - Hugo Pfoertner, Mar 04 2019

References

• Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, pp. 134, 138, 139, 219, answer to exercise 172, Addison-Wesley, 2009.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-6).

Crossrefs

Cf. A083314.

Programs

Magma

[(2*3^n-2^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Feb 01 2015

Maple

A083313 := proc(n)
    if n = 0 then
        1;
    else
        3^n-2^(n-1) ;
    end if;
end proc: # R. J. Mathar, Aug 01 2013

Mathematica

CoefficientList[Series[((1 - x) + (1 - 2 x) (1 - 3 x)) / (2 (1 - 2 x) (1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 01 2015 *)
LinearRecurrence[{5,-6},{1,2,7},30] (* Harvey P. Dale, Sep 04 2017 *)

PARI

Vec(((1-x)+(1-2*x)*(1-3*x))/(2*(1-2*x)*(1-3*x)) + O(x^30)) \\ Michel Marcus, Jan 31 2015

PARI

print1(1,", ",s=2,", " );for(k=2,27,s=2^(k-2)+3*s;print1(s,", ")) \\ Hugo Pfoertner, Mar 04 2019

Formula

a(n) = (2*3^n - (2^n - 0^n))/2.

a(0) = 1, a(n) = 3^n - 2^(n-1) for n >= 1.

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

E.g.f.: (2*exp(3*x) - exp(2*x) + exp(0))/2.

a(n) = A090888(n-1, 4), for n > 0. - Ross La Haye, Sep 21 2004

Let b(n) = 2*(3/2)^n - 1. Then A003063(n) = -b(1-n)*3^(n-1) for n > 0. a(n) = A064686(n) = b(n)*2^(n-1) for n > 0. - Michael Somos, Aug 06 2006

From Alex Ratushnyak, Jul 03 2012: (Start)

a(n) mod 100 = 23 for n = 4*k-1, k >= 1.

a(n) mod 100 = 27 for n = 4*k+1, k >= 1.

(End)

Extensions

Better name by Alex Ratushnyak, Jul 02 2012