OFFSET
0,1
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).
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
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 *)
PROG
(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}
(Magma) [4*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011
(SageMath) [4^(n+1) -2^n for n in range(31)] # G. C. Greubel, Aug 31 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Dec 10 2009
STATUS
approved