A003674 a(n) = 2^(n-1)*(2^n - (-1)^n).

0, 3, 6, 36, 120, 528, 2016, 8256, 32640, 131328, 523776, 2098176, 8386560, 33558528, 134209536, 536887296, 2147450880, 8590000128, 34359607296, 137439215616, 549755289600, 2199024304128, 8796090925056

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,8).

Crossrefs

Cf. A001045, A003683 (one-third), A062510, A071930.

Programs

Magma

[(4^n -(-2)^n)/2: n in [0..40]]; // G. C. Greubel, Feb 17 2023

Mathematica

Table[(4^n-(-2)^n)/2, {n,0,40}] (* G. C. Greubel, Feb 17 2023 *)

PARI

a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n))

SageMath

[(4^n-(-2)^n)/2 for n in range(41)] # G. C. Greubel, Feb 17 2023

Formula

G.f.: 3*x/((1+2*x)*(1-4*x)).

a(n) = 3*A003683(n).

Given the 2 X 2 matrix M = [1,3; 3,1], a(n) = term (1,2) in M^n, n>0. - Gary W. Adamson, Aug 06 2010

From G. C. Greubel, Feb 17 2023: (Start)

a(n) = 2*a(n-1) + 8*a(n-2).

a(n) = 3*2^(n-1)*A001045(n).

a(n) = 2^(n-1)*A062510(n).

a(n) = (1/2)*A071930(n+1).

E.g.f.: (1/2)*(exp(4*x) - exp(-2*x)). (End)