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%I #20 Sep 08 2022 08:45:09
%S 1,7,34,148,616,2512,10144,40768,163456,654592,2619904,10482688,
%T 41936896,167759872,671064064,2684305408,10737319936,42949476352,
%U 171798298624,687193980928,2748777496576,10995113132032,43980458819584
%N Third binomial transform of A010685 (period 2: repeat 1,4).
%H Vincenzo Librandi, <a href="/A080960/b080960.txt">Table of n, a(n) for n = 0..300</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F a(n) = 4*a(n-1) + 3*2^(n-1).
%F a(n) = (5/2)*4^n - (3/2)*2^n.
%F G.f.: (1+x)/((1-2*x)*(1-4*x)). - _Klaus Brockhaus_, Nov 26 2009
%F a(n) = 6*a(n-1) - 8*a(n-2), a(0)=1, a(1)=7. - _Harvey P. Dale_, Nov 12 2012
%F E.g.f.: exp(2*x)*(5*exp(2*x) - 3)/2. - _G. C. Greubel_, Nov 23 2021
%t RecurrenceTable[{a[0]==1,a[n]==4a[n-1]+3*2^(n-1)},a,{n,30}] (* or *) LinearRecurrence[{6,-8},{1,7},30] (* _Harvey P. Dale_, Nov 12 2012 *)
%t CoefficientList[Series[(1+x)/((1-2x)(1-4x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%o (Magma) binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >; binomtf(binomtf(binomtf(&cat[ [1, 4]: n in [1..12] ]))); // _Klaus Brockhaus_, Nov 26 2009
%o (Sage) [2^(n-1)*(5*2^n -3) for n in (0..30)] # _G. C. Greubel_, Nov 23 2021
%Y Cf. A003945, A010685, A048473, A080961.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 03 2003
%E Definition corrected, edited by _Klaus Brockhaus_, Nov 26 2009
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