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%I #37 Sep 08 2022 08:45:50
%S 1,6,28,120,496,2016,8128,32640,130816,523776,2096128,8386560,
%T 33550336,134209536,536854528,2147450880,8589869056,34359607296,
%U 137438691328,549755289600,2199022206976,8796090925056,35184367894528
%N a(n) = 6*a(n-1) - 8*a(n-2) for n > 1, a(0)=1, a(1)=6.
%C Binomial transform of A048473; second binomial transform of A151821; third binomial transform of A010684; fourth binomial transform of A084633 without second term 0; fifth binomial transform of A168589.
%C Inverse binomial transform of A081625; second inverse binomial transform of A081626; third inverse binomial transform of A081627.
%C Partial sums of A010036.
%C Essentially first differences of A006095.
%C a(n) = A109241(n) converted from binary to decimal. - _Robert Price_, Jan 19 2016
%H Vincenzo Librandi, <a href="/A171476/b171476.txt">Table of n, a(n) for n = 0..1000</a>
%H Gordon Hamilton, <a href="http://www.youtube.com/watch?v=8xwCHeI7Vu4">Cookie Monster Counts Cookies</a>, Youtube video, 2010.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F a(n) = Sum_{k=1..2^n-1} k.
%F a(n) = 2*4^n - 2^n.
%F G.f.: 1/((1-2*x)*(1-4*x)).
%F a(n) = A006516(n+1).
%F a(n) = 4*a(n-1) + 2^n for n > 0, a(0)=1. - _Vincenzo Librandi_, Jul 17 2011
%F a(n) = Sum_{k=0..n} 2^(n+k). - _Bruno Berselli_, Aug 07 2013
%F a(n) = A020522(n+1)/2. - _Hussam al-Homsi_, Jun 06 2021
%F E.g.f.: exp(2*x)*(2*exp(2*x) - 1). - _Stefano Spezia_, Dec 10 2021
%t LinearRecurrence[{6,-8},{1,6},30] (* _Harvey P. Dale_, Aug 02 2020 *)
%o (PARI) m=23; v=concat([1, 6], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v
%o (Magma) [2*4^n-2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011
%Y Cf. A006516 (2^(n-1)*(2^n-1)), A020522 (4^n-2^n), A048473 (2*3^n-1), A151821 (powers of 2, omitting 2 itself), A010684 (repeat 1, 3), A084633 (inverse binomial transform of repeated odd numbers), A168589 ((2-3^n)*(-1)^n), A081625 (2*5^n-3^n), A081626 (2*6^n-4^n), A081627 (2*7^n-5^n), A010036 (sum of 2^n, ..., 2^(n+1)-1), A006095 (Gaussian binomial coefficient [n, 2] for q=2), A171472, A171473.
%K nonn,easy
%O 0,2
%A _Klaus Brockhaus_, Dec 09 2009
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