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

%I A171499 #53 Aug 31 2023 03:58:41
%S A171499 3,14,60,248,1008,4064,16320,65408,261888,1048064,4193280,16775168,
%T A171499 67104768,268427264,1073725440,4294934528,17179803648,68719345664,
%U A171499 274877644800,1099511103488,4398045462528,17592183947264,70368739983360
%N A171499 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 3, a(1) = 14.
%C A171499 Binomial transform of A171498; second binomial transform of A171497; third binomial transform of A010703.
%C A171499 Related to sequences A001969 and A000069, sum of each group with exponent 1. - _Eric Desbiaux_, Jul 24 2013
%C A171499 a(n) in base 2 is n+2 1s followed by n 0s. - _Hussam al-Homsi_, Oct 12 2021
%H A171499 Vincenzo Librandi, <a href="/A171499/b171499.txt">Table of n, a(n) for n = 0..500</a>
%H A171499 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F A171499 a(n) = 4*4^n - 2^n = 2^n * (2^(n+2) - 1).
%F A171499 G.f.: (3-4*x)/((1-2*x)*(1-4*x)).
%F A171499 a(n) = 4*a(n-1) + 2^n for n > 0. - _Vincenzo Librandi_, Jul 18 2011
%F A171499 a(n) = A171476(n+1)/2. - _Hussam al-Homsi_, Jun 06 2021
%F A171499 E.g.f.: 4*exp(4*x) - exp(2*x). - _G. C. Greubel_, Aug 31 2023
%t A171499 (* This program shows how A171499 arises from the Vandermonde determinant of (1,2,4,...,2^(n-1)). *)
%t A171499 f[j_]:= 2^j - 1; z = 15;
%t A171499 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A171499 d[n_]:= Product[(i-1)!, {i,n}]
%t A171499 Table[v[n], {n,z}]                     (* A203303 *)
%t A171499 Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
%t A171499 Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *)
%t A171499 (* _Clark Kimberling_, Jan 02 2011 *)
%t A171499 LinearRecurrence[{6,-8},{3,14},30] (* _Harvey P. Dale_, Sep 05 2021 *)
%o A171499 (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 A171499 (Magma) [4*4^n-2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 18 2011
%o A171499 (SageMath) [4^(n+1) -2^n for n in range(31)] # _G. C. Greubel_, Aug 31 2023
%Y A171499 Cf. A010703, A171472, A171473, A171476, A171497, A171498.
%K A171499 nonn,easy
%O A171499 0,1
%A A171499 _Klaus Brockhaus_, Dec 10 2009