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

3, 8, 17, 34, 67, 132, 261, 518, 1031, 2056, 4105, 8202, 16395, 32780, 65549, 131086, 262159, 524304, 1048593, 2097170, 4194323, 8388628, 16777237, 33554454, 67108887, 134217752, 268435481, 536870938, 1073741851, 2147483676, 4294967325, 8589934622, 17179869215

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

Keywords

• nonn
• easy

Comments

The third number that is a sum of n positive n-th powers. - Alois P. Heinz, Aug 02 2020

Examples

			a(2) = a(1) + 2^2 + 1 = 3 + 4 + 1 = 8; a(3) = a(2) + 2^3 + 1 = 8 + 8 + 1 = 17.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000051 (2^n + 1), A000225 (2^n - 1), A000295 (Eulerian numbers).

Cf. A052944, A127330.

Column k = 1 of triangle A308737.

Row n=3 of A336725.

Programs

ARIBAS

a:=0; for n:=1 to 30 do a:=a+2**n+1; write(a,","); end;

Haskell

a145071 n = 2 ^ (n + 1) + n - 2
a145071_list = scanl1 (+) $ tail a000051_list
-- Reinhard Zumkeller, Nov 16 2013

Mathematica

lst={};s=0;Do[s+=2^n+1;AppendTo[lst,s],{n,5!}];lst
Accumulate[2^Range[30]+1] (* Harvey P. Dale, Feb 19 2023 *)

Formula

a(1) = 3; a(n) = a(n-1) + 2^n + 1 for n > 1.

a(n) = 2^(n+1) + n - 2. - Franklin T. Adams-Watters, Jul 06 2009

G.f.: x*(3-4*x)/((1-x)^2*(1-2*x)). - Colin Barker, Jan 11 2012

a(n) = A127330(n,n) = A052944(n-1) + 2. - Reinhard Zumkeller, Nov 16 2013

From Elmo R. Oliveira, Apr 01 2025: (Start)

E.g.f.: exp(x)*(x - 2 + 2*exp(x)).

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3. (End)

Extensions

Edited by Klaus Brockhaus, Oct 14 2008