|
%I #70 Apr 15 2023 06:27:21
%S 1,3,6,10,15,21,28,36,45,65,87,111,137,165,195,227,261,297,335,375,
%T 417,461,507,555,605,657,711,767,825,885,947,1011,1077,1145,1215,1287,
%U 1361,1437,1515,1595,1677,1761,1847,1935,2025,2117,2211,2307,2405,2505,2607,2711,2817,2925
%N a(n) is the number of decimal digits in A098129(n) and A300517(n).
%H Winston de Greef, <a href="/A361751/b361751.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A055642(A098129(n)).
%F From _Alois P. Heinz_, Mar 23 2023: (Start)
%F a(n) = Sum_{j=1..n} j*A055642(j).
%F a(n) = Sum_{j=1..n} A110803(j). (End)
%F a(n) = Sum_{k=0..floor(log_10(n))} (n*(n+1) - 10^k*(10^k-1))/2. - _Andrew Howroyd_, Mar 24 2023
%F a(n) = k*n*(n+1)/2 - ((100^k-1)/99 - (10^k-1)/9)/2, where k = floor(log_10(n))+1. - _David Cleaver_, Mar 25 2023
%e For n = 4, a(4) = 10, because A098129(4) = 1223334444.
%e For n = 10, a(10) = 65, because A098129(10) = 12233344445555566666677777778888888899999999910101010101010101010.
%p a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+n*length(n)) end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Mar 23 2023
%o (PARI)
%o a(n) = {my(x=logint(n,10)+1);x*n*(n+1)/2 - ((100^x-1)/99 - (10^x-1)/9)/2}
%o vector(100, i, a(i))
%o (Python)
%o def a(n):
%o d = len(str(n))
%o m = 10**d
%o return d*n*(n+1)//2 - ((m-11)*m + 10)//198
%o print([a(n) for n in range(1, 55)]) # _Michael S. Branicky_, Mar 24 2023 modified Mar 29 2023
%o (Python) # faster for generating initial segment of sequence
%o from itertools import count, islice
%o def agen(s=0): yield from (s:=s+n*len(str(n)) for n in count(1))
%o print(list(islice(agen(), 60))) # _Michael S. Branicky_, Mar 24 2023
%Y Cf. A055642, A098129, A300517.
%Y Partial sums of A110803.
%K nonn,base,easy
%O 1,2
%A _David Cleaver_, Mar 23 2023
|
