A171499 - OEIS

%I #53 Aug 31 2023 03:58:41

%S 3,14,60,248,1008,4064,16320,65408,261888,1048064,4193280,16775168,

%T 67104768,268427264,1073725440,4294934528,17179803648,68719345664,

%U 274877644800,1099511103488,4398045462528,17592183947264,70368739983360

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

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

%C Related to sequences A001969 and A000069, sum of each group with exponent 1. - _Eric Desbiaux_, Jul 24 2013

%C a(n) in base 2 is n+2 1s followed by n 0s. - _Hussam al-Homsi_, Oct 12 2021

%H Vincenzo Librandi, <a href="/A171499/b171499.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).

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

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

%F a(n) = 4*a(n-1) + 2^n for n > 0. - _Vincenzo Librandi_, Jul 18 2011

%F a(n) = A171476(n+1)/2. - _Hussam al-Homsi_, Jun 06 2021

%F E.g.f.: 4*exp(4*x) - exp(2*x). - _G. C. Greubel_, Aug 31 2023

%t (* This program shows how A171499 arises from the Vandermonde determinant of (1,2,4,...,2^(n-1)). *)

%t f[j_]:= 2^j - 1; z = 15;

%t v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]

%t d[n_]:= Product[(i-1)!, {i,n}]

%t Table[v[n], {n,z}]                     (* A203303 *)

%t Table[v[n+1]/v[n], {n,z}]              (* A002884 *)

%t Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *)

%t (* _Clark Kimberling_, Jan 02 2011 *)

%t LinearRecurrence[{6,-8},{3,14},30] (* _Harvey P. Dale_, Sep 05 2021 *)

%o (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}

%o (Magma) [4*4^n-2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 18 2011

%o (SageMath) [4^(n+1) -2^n for n in range(31)] # _G. C. Greubel_, Aug 31 2023

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

%K nonn,easy

%O 0,1

%A _Klaus Brockhaus_, Dec 10 2009