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A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

%I A340207 #12 May 06 2022 13:13:51
%S A340207 9,8,1,9,6,1,9,8,0,1,9,9,8,5,6,9,9,8,0,0,1,9,9,9,8,2,4,4,9,9,9,8,0,0,
%T A340207 0,1,9,9,9,9,5,0,8,8,4,9,9,9,9,8,0,0,0,0,1,9,9,9,9,9,5,1,5,5,2,9,9,9,
%U A340207 9,9,9,8,0,0,0,0,0,1,9,9,9,9,9,9,5,8
%N A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...
%C A340207 The terms of sequence A339978 converge to this sequence of digits, and to this constant, up to powers of 10.
%H A340207 Harvey P. Dale, <a href="/A340207/b340207.txt">Table of n, a(n) for n = 0..1000</a>
%F A340207 c = 0.9819619801998569980019998244999800019999508849999800001999995155...
%F A340207   = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/2)-1)^2
%F A340207 a(-n(n+1)/2) = 9 for all n >= 2.
%e A340207 The largest square with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 81 = 9^2, 961 = 31^2, 9801 = 99^2, ....
%e A340207 Here we list the sequence of digits of these numbers: 9; 8, 1; 9, 6, 1; 9, 8, 0, 1; 9, 9, 8, 5, 6; ...
%e A340207 This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.98196198...
%t A340207 lnds[k_]:=Module[{c=Sqrt[10^k]},If[IntegerQ[c],(c-1)^2,Floor[c]^2]]; Flatten[IntegerDigits/@(lnds/@Range[15])] (* _Harvey P. Dale_, Dec 16 2021 *)
%o A340207 (PARI) concat([digits(sqrtint(10^k-1)^2)|k<-[1..14]]) \\ as seq. of digits
%o A340207 c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtint(10^k-1)^2) \\ as constant
%Y A340207 Cf. A061433 (largest n-digit square), A339978 (has this as "limit"), A340208 (same with "smallest n-digit cube", limit of A215692), A340209 (same for cubes, limit of A340115), A340220 (same for primes), A340219 (similar, with smallest primes, limit of A215641), A340222 (same for semiprimes), A340221 (same for smallest semiprimes, limit of A215647).
%Y A340207 Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).
%K A340207 nonn,base,cons
%O A340207 0,1
%A A340207 _M. F. Hasler_, Jan 01 2021