cp's OEIS Frontend

A190730 Let b(n,0) = n and b(n,k) = 2*b(n,k-1) + 1 for k > 0. Then a(n) = b(n,1) + b(n,2) + ... + b(n,n).

%I A190730 #45 Oct 17 2023 05:08:40
%S A190730 3,16,53,146,367,876,2025,4582,10211,22496,49117,106458,229335,491476,
%T A190730 1048529,2228174,4718539,9961416,20971461,44040130,92274623,192937916,
%U A190730 402653113,838860726,1744830387,3623878576,7516192685,15569256362,32212254631,66571992996
%N A190730 Let b(n,0) = n and b(n,k) = 2*b(n,k-1) + 1 for k > 0.  Then a(n) = b(n,1) + b(n,2) + ... + b(n,n).
%C A190730 It turns out that b(n,k) = A087322(n,k) = (n + 1)*2^k - 1 for 1 <= k <= n (without the 0th column). - _Petros Hadjicostas_, Feb 15 2021
%H A190730 Vincenzo Librandi, <a href="/A190730/b190730.txt">Table of n, a(n) for n = 1..1000</a>
%H A190730 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).
%F A190730 a(n) = (n+1) * 2^(n+1) - 3*n - 2 = A036289(n+1) - A016789(n).
%F A190730 G.f.: -x*(-3 + 2*x + 4*x^2) / ( (2*x-1)^2*(x-1)^2 ). - _R. J. Mathar_, May 29 2011
%F A190730 E.g.f.: exp(x)*(2*exp(x)*(1 + 2*x) - 2 - 3*x). - _Stefano Spezia_, Oct 16 2023
%e A190730 One way to view it is to begin with n = 5, then 5 + 6 = 11 --> 11 + 12 = 23 --> 23 + 24 = 47 --> 47 + 48 = 95 --> 95 + 96 = 191. There are n steps, in this case 5, that give the sum 11 + 23 + 47 + 95 + 191 = 367. This is the same as (2*5+1) + (4*5+3) + (8*5+7) + (16*5+15) + (32*5+31). The formula gives (5+1)*2^(5+1) - 3*5 - 2 = 6*64 - 17 = 367.
%t A190730 LinearRecurrence[{6,-13,12,-4},{3,16,53,146},40] (* or *)
%t A190730 Array[(#+1)2^(#+1)-3#-2&,40] (* _Paolo Xausa_, Oct 17 2023 *)
%o A190730 (Magma) [(n+1) * 2^(n+1) - 3*n - 2 : n in [1..30]]; // _Vincenzo Librandi_, Sep 29 2011
%Y A190730 Cf. A016789, A036289, A087322.
%K A190730 nonn,easy
%O A190730 1,1
%A A190730 _J. M. Bergot_, May 17 2011