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%I #45 Sep 08 2022 08:45:17
%S 3,18,84,360,1488,6048,24384,97920,392448,1571328,6288384,25159680,
%T 100651008,402628608,1610563584,6442352640,25769607168,103078821888,
%U 412316073984,1649265868800,6597066620928,26388272775168,105553103683584,422212439900160
%N a(n) = 3*2^(n-1)*(2^n-1).
%C Divide the sequence of natural numbers: s0=1,2,3,4,5,6,7,8,9,10,11,12,13,14,... into sections s(n) of length 2*s1-1, where s1=initial digits of s(n): s={1,2},{3,4,5,6},{7,8,9,10,11,12,13,14},... then a(n)=sum of terms of s(n): 3,18,84,...
%C Sum of the numbers between 2^n and 2^(n+1), both excluded. - _Gionata Neri_, Jun 16 2015
%H Vincenzo Librandi, <a href="/A103897/b103897.txt">Table of n, a(n) for n = 1..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F a(n) = 3*A006516(n).
%F From _Bruno Berselli_, Sep 19 2011: (Start)
%F G.f.: 3*x/((1-2*x)*(1-4*x)).
%F a(n+2) = A061561(4n-2). (End)
%F E.g.f.: (3/2)*(exp(4*x) - exp(2*x)). - _Stefano Spezia_, Nov 10 2019
%t Table[3*2^(n - 1)*(2^n - 1), {n, 30}]
%t LinearRecurrence[{6,-8},{3,18},30] (* _Harvey P. Dale_, Feb 11 2018 *)
%o (Magma) [3*2^(n-1)*(2^n-1): n in [1..24]]; // _Bruno Berselli_, Sep 19 2011
%o (PARI) a(n)=3*2^(n-1)*(2^n-1) \\ _Charles R Greathouse IV_, Jun 08 2015
%o (Python) b = list(range(0,2**20-1)); a = [sum(b[2**i-1:2**(i+1)-1]) for i in range(1,20)] ## _Johan Claes_, Nov 10 2019
%Y Cf. A006516.
%K nonn,easy
%O 1,1
%A _Zak Seidov_, Mar 30 2005
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