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A145071 Partial sums of A000051, starting at n=1.

%I A145071 #33 Apr 02 2025 04:15:39
%S A145071 3,8,17,34,67,132,261,518,1031,2056,4105,8202,16395,32780,65549,
%T A145071 131086,262159,524304,1048593,2097170,4194323,8388628,16777237,
%U A145071 33554454,67108887,134217752,268435481,536870938,1073741851,2147483676,4294967325,8589934622,17179869215
%N A145071 Partial sums of A000051, starting at n=1.
%C A145071 The third number that is a sum of n positive n-th powers. - _Alois P. Heinz_, Aug 02 2020
%H A145071 Vincenzo Librandi, <a href="/A145071/b145071.txt">Table of n, a(n) for n = 1..1000</a>
%H A145071 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A145071 a(1) = 3; a(n) = a(n-1) + 2^n + 1 for n > 1.
%F A145071 a(n) = 2^(n+1) + n - 2. - _Franklin T. Adams-Watters_, Jul 06 2009
%F A145071 G.f.: x*(3-4*x)/((1-x)^2*(1-2*x)). - _Colin Barker_, Jan 11 2012
%F A145071 a(n) = A127330(n,n) = A052944(n-1) + 2. - _Reinhard Zumkeller_, Nov 16 2013
%F A145071 From _Elmo R. Oliveira_, Apr 01 2025: (Start)
%F A145071 E.g.f.: exp(x)*(x - 2 + 2*exp(x)).
%F A145071 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3. (End)
%e A145071 a(2) = a(1) + 2^2 + 1 = 3 + 4 + 1 = 8; a(3) = a(2) + 2^3 + 1 = 8 + 8 + 1 = 17.
%t A145071 lst={};s=0;Do[s+=2^n+1;AppendTo[lst,s],{n,5!}];lst
%t A145071 Accumulate[2^Range[30]+1] (* _Harvey P. Dale_, Feb 19 2023 *)
%o A145071 (ARIBAS) a:=0; for n:=1 to 30 do a:=a+2**n+1; write(a,","); end;
%o A145071 (Haskell)
%o A145071 a145071 n = 2 ^ (n + 1) + n - 2
%o A145071 a145071_list = scanl1 (+) $ tail a000051_list
%o A145071 -- _Reinhard Zumkeller_, Nov 16 2013
%Y A145071 Cf. A000051 (2^n + 1), A000225 (2^n - 1), A000295 (Eulerian numbers).
%Y A145071 Cf. A052944, A127330.
%Y A145071 Column k = 1 of triangle A308737.
%Y A145071 Row n=3 of A336725.
%K A145071 nonn,easy
%O A145071 1,1
%A A145071 _Vladimir Joseph Stephan Orlovsky_, Sep 30 2008
%E A145071 Edited by _Klaus Brockhaus_, Oct 14 2008