|
| |
|
|
A340206
|
|
Constant whose decimal expansion is the concatenation of the smallest n-digit square A061432(n), for n = 1, 2, 3, ...
|
|
5
|
|
|
|
1, 1, 6, 1, 0, 0, 1, 0, 2, 4, 1, 0, 0, 0, 0, 1, 0, 0, 4, 8, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 5, 6, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 4, 1, 2, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 7, 9, 8, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 4, 9, 2, 8, 4, 1, 0, 0, 0, 0
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
The terms of sequence A215689 have this as limit, digit-wise and as a constant, up to powers of 10.
Every other "smallest n-digit square" (i.e., for odd n = 2k + 1) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.
|
|
|
LINKS
|
|
|
|
FORMULA
|
c = 0.11610010241000010048910000001000456910000000010000141291000000000010...
= Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/2))^2
a(-n(n+1)/2) = 1 for all n >= 0; a(k) = 0 for -n(n-1)/2 > k > -n(n+1)/2 with odd n.
|
|
|
EXAMPLE
|
The smallest square with 1, 2, 3, 4, ... digits is, respectively, 1, 16 = 4^2, 100 = 10^2, 1024 = 32^2, ....
Here we list the digits of these numbers: 1; 1, 6; 1, 0, 0; 1, 0, 2, 4; ...
As for the Champernowne and Copeland-Erdős constants, we can consider this as the decimal expansion of the real constant 0.116100102410000100489...
|
|
|
PROG
|
(PARI) concat([digits(ceil(10^((k-1)/2))^2)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1, N, .1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
