A171499 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 3, a(1) = 14.

3, 14, 60, 248, 1008, 4064, 16320, 65408, 261888, 1048064, 4193280, 16775168, 67104768, 268427264, 1073725440, 4294934528, 17179803648, 68719345664, 274877644800, 1099511103488, 4398045462528, 17592183947264, 70368739983360

Offset: 0

Author

Klaus Brockhaus, Dec 10 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A171498; second binomial transform of A171497; third binomial transform of A010703.

Related to sequences A001969 and A000069, sum of each group with exponent 1. - Eric Desbiaux, Jul 24 2013

a(n) in base 2 is n+2 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (6,-8).

Crossrefs

Cf. A010703, A171472, A171473, A171476, A171497, A171498.

Programs

Magma

[4*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

Mathematica

(* This program shows how A171499 arises from the Vandermonde determinant of (1,2,4,...,2^(n-1)). *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]                     (* A203303 *)
Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *)
(* Clark Kimberling, Jan 02 2011 *)
LinearRecurrence[{6,-8},{3,14},30] (* Harvey P. Dale, Sep 05 2021 *)

PARI

{m=23; v=concat([3, 14], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

SageMath

[4^(n+1) -2^n for n in range(31)] # G. C. Greubel, Aug 31 2023

Formula

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

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

a(n) = 4*a(n-1) + 2^n for n > 0. - Vincenzo Librandi, Jul 18 2011

a(n) = A171476(n+1)/2. - Hussam al-Homsi, Jun 06 2021

E.g.f.: 4*exp(4*x) - exp(2*x). - G. C. Greubel, Aug 31 2023