A031973 a(n) = Sum_{k=0..n} n^k.

1, 2, 7, 40, 341, 3906, 55987, 960800, 19173961, 435848050, 11111111111, 313842837672, 9726655034461, 328114698808274, 11966776581370171, 469172025408063616, 19676527011956855057, 878942778254232811938, 41660902667961039785743, 2088331858752553232964200

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

These are the generalized repunits of length n+1 in base n for all n >= 1: a(n) expressed in base n is 111...111 (n+1 1's): a(1) = 1^0 + 1^1 = 2 = A000042(2), a(2) = 2^0 + 2^1 + 2^2 = 7 = A000225(3), a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40 = A003462(4), etc., a(10) = 10^0 + 10^1 + 10^2 + ... + 10^9 + 10^10 = 11111111111 = A002275(11), etc. - Rick L. Shepherd, Aug 26 2004

a(n)=the total number of ordered selections of up to n objects from n types with repetitions allowed. Thus for 2 objects a,b there are 7 possible selections: aa,bb,ab,ba,a,b, and the null set. - J. M. Bergot, Mar 26 2014

a(n)=the total number of ordered arrangements of 0,1,2..n objects, with repetitions allowed, selected from n types of objects. - J. M. Bergot, Apr 11 2014

Examples

			a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000042 (unary representations), A000225 (2^n-1: binary repunits shown in decimal), A003462 ((3^n-1)/2: ternary repunits shown in decimal), A002275 ((10^n-1)/9: decimal repunits).

Cf. A104878.

Programs

Magma

[&+[n^k: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

Maple

a:= proc(n) local c, i; c:=1; for i to n do c:= c*n+1 od; c end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 15 2013

Mathematica

Join[{1},Table[Total[n^Range[0,n]],{n,20}]] (* Harvey P. Dale, Nov 13 2011 *)

PARI

a(n)=(n^(n+1)-1)/(n-1) \\ Charles R Greathouse IV, Mar 26 2014

Sage

[lucas_number1(n,n,n-1) for n in range(1, 19)] # Zerinvary Lajos, May 16 2009

Formula

a(n) = (n^(n+1)-1)/(n-1) = (A007778(n)-1)/(n-1) = A023037(n)+A000312(n) = A031972(n)+1. - Henry Bottomley, Apr 04 2003

a(n) = A125118(n,n-2) for n>2. - Reinhard Zumkeller, Nov 21 2006

a(n) = [x^n] 1/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017

a(n) = A104878(2n,n). - Alois P. Heinz, May 04 2021