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%I #29 Sep 08 2022 08:45:50
%S 1,8,42,190,806,3318,13462,54230,217686,872278,3492182,13974870,
%T 55911766,223671638,894735702,3579041110,14316361046,57265837398,
%U 229064136022,916258116950,3665035613526,14660148745558,58640607565142
%N a(n) = 6*a(n-1) - 8*a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 8.
%C Second binomial transform of A168648.
%C Partial sums of A080960.
%H Harvey P. Dale, <a href="/A171478/b171478.txt">Table of n, a(n) for n = 0..1000</a> (* extending prior file from Vincenzo Librandi *)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7, -14, 8).
%F a(n) = (10*4^n - 9*2^n + 2)/3.
%F G.f.: (1+x)/((1-x)*(1-2*x)*(1-4*x)).
%F a(0)=1, a(1)=8, a(2)=42, a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - _Harvey P. Dale_, May 04 2012
%F a(n) = A203241(n+1) + 2^n*(2^(n+1)-1), n>0. - _J. M. Bergot_, Mar 21 2018
%p a:= proc(n) option remember: if n = 0 then 1 elif n = 1 then 8 elif n >= 2 then 6*procname(n-1) - 8*procname(n-2) + 2 fi; end:
%p seq(a(n), n = 0..25); # _Muniru A Asiru_, Mar 22 2018
%t RecurrenceTable[{a[0]==1,a[1]==8,a[n]==6a[n-1]-8a[n-2]+2},a,{n,30}] (* or *) LinearRecurrence[{7,-14,8},{1,8,42},30] (* _Harvey P. Dale_, May 04 2012 *)
%o (PARI) {m=23; v=concat([1, 8], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]+2); v}
%o (Magma) [(10*4^n-9*2^n+2)/3: n in [0..30]]; // _Vincenzo Librandi_, Jul 18 2011
%o (GAP) a:=[1,8];; for n in [3..25] do a[n]:=6*a[n-1]-8*a[n-2]+2; od; a; # _Muniru A Asiru_, Mar 22 2018
%Y Cf. A168648 ((10*2^n+2*(-1)^n)/3, a(0)=1), A080960 (third binomial transform of A010685), A171472, A171473.
%K nonn,easy
%O 0,2
%A _Klaus Brockhaus_, Dec 09 2009
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