A361751 a(n) is the number of decimal digits in A098129(n) and A300517(n).
1, 3, 6, 10, 15, 21, 28, 36, 45, 65, 87, 111, 137, 165, 195, 227, 261, 297, 335, 375, 417, 461, 507, 555, 605, 657, 711, 767, 825, 885, 947, 1011, 1077, 1145, 1215, 1287, 1361, 1437, 1515, 1595, 1677, 1761, 1847, 1935, 2025, 2117, 2211, 2307, 2405, 2505, 2607, 2711, 2817, 2925
Offset: 1
Examples
For n = 4, a(4) = 10, because A098129(4) = 1223334444. For n = 10, a(10) = 65, because A098129(10) = 12233344445555566666677777778888888899999999910101010101010101010.
Links
- Winston de Greef, Table of n, a(n) for n = 1..10000
Programs
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Maple
a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+n*length(n)) end: seq(a(n), n=1..100); # Alois P. Heinz, Mar 23 2023
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PARI
a(n) = {my(x=logint(n,10)+1);x*n*(n+1)/2 - ((100^x-1)/99 - (10^x-1)/9)/2} vector(100, i, a(i)) -
Python
def a(n): d = len(str(n)) m = 10**d return d*n*(n+1)//2 - ((m-11)*m + 10)//198 print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Mar 24 2023 modified Mar 29 2023 -
Python
# faster for generating initial segment of sequence from itertools import count, islice def agen(s=0): yield from (s:=s+n*len(str(n)) for n in count(1)) print(list(islice(agen(), 60))) # Michael S. Branicky, Mar 24 2023
Formula
From Alois P. Heinz, Mar 23 2023: (Start)
a(n) = Sum_{j=1..n} j*A055642(j).
a(n) = Sum_{j=1..n} A110803(j). (End)
a(n) = Sum_{k=0..floor(log_10(n))} (n*(n+1) - 10^k*(10^k-1))/2. - Andrew Howroyd, Mar 24 2023
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
Comments