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A096978 Sum of the areas of the first n Jacobsthal rectangles.

%I A096978 #17 Sep 08 2022 08:45:14
%S A096978 0,1,4,19,74,305,1208,4863,19398,77709,310612,1242907,4970722,
%T A096978 19884713,79535216,318148151,1272578046,5090341317,20361307020,
%U A096978 81445344595,325781145370,1303125047521,5212499258024,20849998896239,83399991856694
%N A096978 Sum of the areas of the first n Jacobsthal rectangles.
%H A096978 Vincenzo Librandi, <a href="/A096978/b096978.txt">Table of n, a(n) for n = 0..200</a>
%H A096978 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,3,-14,8).
%F A096978 G.f.: x/((1-x)^2*(1+2*x)*(1-4*x)).
%F A096978 a(n) = 8*4^n/27 - 2*(-2)^n/27 - (n+2)/9;
%F A096978 a(n) = Sum_{k=0..n} (2*4^k/3 + (-2)^k/3)*(n-k).
%F A096978 a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).
%F A096978 a(n) = Sum_{k=0..n} A001045(k)*A001045(k+1).
%F A096978 a(n-1) = Sum_{k=0..n} (-1)^(k+1)*A001045(k)*A001045(2*(n-k)). - _Paul Barry_, Aug 11 2009
%t A096978 LinearRecurrence[{4,3,-14,8},{0,1,4,19},30] (* or *) Table[(2^(2n+1)-3n - 3+(-2)^n)/27,{n,30}] (* _Harvey P. Dale_, Aug 08 2011 *)
%o A096978 (Magma) [8*4^n/27-2*(-2)^n/27-(n+2)/9: n in [0..30]]; // _Vincenzo Librandi_, May 31 2011
%Y A096978 Cf. A096977, A064831.
%K A096978 easy,nonn
%O A096978 0,3
%A A096978 _Paul Barry_, Jul 17 2004