A031972 a(n) = Sum_{k=1..n} n^k.

0, 1, 6, 39, 340, 3905, 55986, 960799, 19173960, 435848049, 11111111110, 313842837671, 9726655034460, 328114698808273, 11966776581370170, 469172025408063615, 19676527011956855056, 878942778254232811937, 41660902667961039785742, 2088331858752553232964199

Offset: 0

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

Sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^n: a(n) = Sum_{k=1..n} k*A228273(n,k). a(2) = 6 = 2+1+1+2: [1,1], [1,2], [2,1], [2,2]. - Alois P. Heinz, Aug 19 2013

a(n) is the expected wait time to see the contiguous subword 11...1 (n copies of 1) over all infinite sequences on alphabet {1,2,...,n}. - Geoffrey Critzer, May 19 2014

a(n) is the number of sequences of k elements from {1,2,...,n}, where 1<=k<=n. For example, a(2) = 6, counting the sequences, [1], [2], [1,1], [1,2], [2,1], [2,2]. Equivalently, a(n) is the number of bar graphs having a height and width of at most n. - Emeric Deutsch, Jan 24 2017.

In base n, a(n) has n+1 digits: n 1's followed by a 0. - Mathew Englander, Oct 20 2020

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386
• A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.

Crossrefs

Main diagonal of A228275.

Cf. A031973, A228273, A023037, A226238.

Programs

Haskell

a031972 n = sum $ take n $ iterate (* n) n
-- Reinhard Zumkeller, Nov 22 2014

Magma

[1] cat [(n^(n+1)-n)/(n-1): n in [2..20]]; // Vincenzo Librandi, Apr 16 2015

Maple

a:= n-> `if`(n<2, n, (n^(n+1)-n)/(n-1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2013

Mathematica

f[n_]:=Sum[n^k,{k,n}];Array[f,30] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

Formula

a(1)=1; for n!=1 a(n) = (n^(n+1)-1)/(n-1) - 1. - Benoit Cloitre, Aug 17 2002

a(n) = A031973(n)-1 for n>0. - Robert G. Wilson v, Apr 15 2015

a(n) = n*A023037(n) = n^n - 1 + A023037(n). - Mathew Englander, Oct 20 2020

Extensions

a(0)=0 prepended by Alois P. Heinz, Oct 22 2019