A226238 -id:A226238 - cp's OEIS frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 14, 12, 4, 0, 0, 5, 30, 39, 20, 5, 0, 0, 6, 62, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 19 2013

A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 14 = 3+1+1+2+2+1+1+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,      0, ...
  0, 1,  2,   3,    4,     5,      6,      7, ...
  0, 2,  6,  14,   30,    62,    126,    254, ...
  0, 3, 12,  39,  120,   363,   1092,   3279, ...
  0, 4, 20,  84,  340,  1364,   5460,  21844, ...
  0, 5, 30, 155,  780,  3905,  19530,  97655, ...
  0, 6, 42, 258, 1554,  9330,  55986, 335922, ...
  0, 7, 56, 399, 2800, 19607, 137256, 960799, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
Main diagonal gives A031972.
Lower diagonal gives A226238.
Cf. A228250.

A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
A(n,k) = Sum_{i=1..k} n^i.
A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - Yosu Yurramendi, Sep 03 2013

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 11 1999

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

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.

Main diagonal of A228275.

Cf. A031973, A228273, A023037, A226238.

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

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

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

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

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

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

A165617 a(n) is the number of positive integers k such that k is equal to the number of 1's in the digits of the base-n expansion of all positive integers <= k.

Original entry on oeis.org

2, 4, 8, 4, 21, 5, 45, 49, 83, 10, 269, 11, 202, 412, 479, 15, 1108, 15, 1545, 1219, 1343, 21, 8944, 706, 1043, 5077, 4084, 28, 27589, 27, 32160, 10423, 6689

Offset: 2

Martin J. Erickson (erickson(AT)truman.edu), Sep 22 2009

The greatest number counted by a(n) is 1...10, where the number of 1's is n-1. - Martin J. Erickson (erickson(AT)truman.edu), Oct 08 2010
These numbers, described in previous comment, 10(2), 110(3), 1110(4), ... expressed in base 10 are: 2, 12, 84, 780, 9330, 137256, 2396744, 48427560, 1111111110, ... - Michel Marcus, Aug 20 2013
The sequence described in the previous two comments is A226238. - Ralf Stephan, Aug 25 2013

			a(3)=4 since there are four values of k such that k is equal to the number of 1's in the digits of the base-3 expansion of all numbers <= k, namely, 1, 4, 5, 12.
From _Jon E. Schoenfield_, Apr 23 2023: (Start)
In the table below, an asterisk appears on each row k at which the cumulative count of 1's in the base-3 expansion of the positive integers 1..k is equal to k:
.
       k      #1's  cume
  ----------  ----  ----
   1 =   1_3    1     1*
   2 =   2_3    0     1
   3 =  10_3    1     2
   4 =  11_3    2     4*
   5 =  12_3    1     5*
   6 =  20_3    0     5
   7 =  21_3    1     6
   8 =  22_3    0     6
   9 = 100_3    1     7
  10 = 101_3    2     9
  11 = 102_3    1    10
  12 = 110_3    2    12*
(End)

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 9.

Cf. A014778, A226238.

nn = 7; Table[c = q = 0; Do[c += DigitCount[i, n, 1]; If[c == i, q++], {i, (#^# - #)/(# - 1) &[n]}]; q, {n, 2, nn}] (* Michael De Vlieger, May 24 2023 *)

a(n) = {my(nmax = (n^n - 1)/(n - 1) - 1, s = 0, nb = 0); for (i=1, nmax, my(digs = digits(i, n)); s += sum (k=1, #digs, (digs[k] == 1)); if (s == i, nb++);); nb;} \\ Michel Marcus, Aug 20 2013; corrected Apr 23 2023

Example corrected by Martin J. Erickson (erickson(AT)truman.edu), Sep 25 2009
Definition and a(10) corrected by Tanya Khovanova, Apr 23 2023
a(11)-a(35) from Gregory Marton, Jul 29 2023

A365097 Smallest k > 1 such that the total number of digits "1" required to write the numbers 1..k in base n is equal to k.

Original entry on oeis.org

2, 4, 25, 181, 421, 3930, 8177, 102772, 199981, 3179142, 5971945, 143610511, 210826981, 4754446846, 8589934561, 222195898593, 396718580701, 13494919482970, 20479999999961, 764527028941797, 1168636602822613, 41826814261329722, 73040694872113105, 2855533828630999398

Offset: 2

Andrew Pope, Aug 21 2023

a(10) = A014778(3), being the smallest term > 1 there.

An upper bound is a(n) <= A226238(n) = u, since the digits of u show there are u 1's in numbers 1..u (in base n). - Kevin Ryde, Sep 28 2023

			For n=2, the first k=2 positive integers are 1 = 1_2 and 2 = 10_2, which have a total of two 1's, so a(2) = 2.
For n=3, the first k=4 positive integers, which are 1_3, 2_3, 10_3, and 11_3, have a total of four 1's, which is equal to k, so a(3) = 4.
For n=4, a total of 25 1's occur in the first k=25 positive integers (they occur in 1_4, 10_4, 11_4, 12_4, 13_4, 21_4, 31_4, 100_4, 101_4, 102_4, 103_4, 110_4, 111_4, 112_4, 113_4, 120_4, and 121_4 = 25), so a(4) = 25.

Jon E. Schoenfield, Table of n, a(n) for n = 2..200

Cf. A014778, A094798, A226238.

a[n_] := Module[{k = 1, sum = 1}, While[sum == 1 || sum != k, k++; sum += Count[IntegerDigits[k, n], 1]]; k]; Array[a, 6, 2] (* Amiram Eldar, Aug 29 2023 *)

from itertools import count
from sympy.ntheory.factor_ import digits
def A365097(n):
    c, a, q, m = 1, 1, 0, 1
    for k in count(2):
        m += 1
        if m == n:
            m = 0
            q += 1
            a = digits(q,n).count(1)
        elif m==1:
            a += 1
        elif m==2:
            a -= 1
        c += a
        if c == k:
            return k # Chai Wah Wu, Sep 28 2023

For even n > 2, a(n) = 2*n^(n/2) - 2*n + 1. - Jon E. Schoenfield, Sep 30 2023

a(11)-a(15) from Amiram Eldar, Aug 29 2023
a(16)-a(19) from Chai Wah Wu, Sep 29 2023
a(20)-a(25) from Jon E. Schoenfield, Sep 30 2023

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A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A165617 a(n) is the number of positive integers k such that k is equal to the number of 1's in the digits of the base-n expansion of all positive integers <= k.

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A365097 Smallest k > 1 such that the total number of digits "1" required to write the numbers 1..k in base n is equal to k.

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