

A031972


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


10



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

0,3


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), 3644.


FORMULA

a(0)=0, a(1)=1; for n>1 a(n) = (n^(n+1)1)/(n1)  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


MAPLE

a:= n> `if`(n<2, n, (n^(n+1)n)/(n1)):
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*)


PROG

(Haskell)
a031972 n = sum $ take n $ iterate (* n) n
 Reinhard Zumkeller, Nov 22 2014
(MAGMA) [1] cat [(n^(n+1)n)/(n1): n in [2..20]]; // Vincenzo Librandi, Apr 16 2015


CROSSREFS

Main diagonal of A228275.
Cf. A031973, A228273, A023037, A226238.
Sequence in context: A113347 A265953 A246571 * A308861 A124577 A006678
Adjacent sequences: A031969 A031970 A031971 * A031973 A031974 A031975


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Dec 11 1999


EXTENSIONS

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


STATUS

approved



