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

%I #21 Jul 30 2015 17:02:14

%S 4,56,496,4064,32704,262016,2096896,16776704,134216704,1073739776,

%T 8589930496,68719468544,549755797504,4398046478336,35184372023296,

%U 281474976579584,2251799813423104,18014398508957696,144115188074807296

%N a(n) = 8^(n+1) - 2^(n+2).

%C Third Chebyshev polynomial of second kind evaluated at 2^n.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -16).

%F G.f.: 4(1+4x)/(1-10x+16x^2).

%F a(0)=4, a(1)=56, a(n) = 10*a(n-1) - 16*a(n-2). - _Harvey P. Dale_, Feb 27 2013

%e U_3(x) = 8x^3 - 4x so U_3(2^n) = 8(2^n)^3 - 4(2^n) = 8^(n+1) - 2^(n+2).

%p with(orthopoly):seq(U(3,2^i),i=0..24);

%t Table[ChebyshevU[3,2^n],{n,1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)

%t Table[8^(n+1)-2^(n+2),{n,0,20}] (* or *) LinearRecurrence[{10,-16},{4,56},20] (* _Harvey P. Dale_, Feb 27 2013 *)

%o (PARI) a(n)=if(n<0,0,8^(n+1)-2^(n+2))

%K nonn

%O 0,1

%A _Simon Plouffe_