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A275623 Integers of the form Sum_{k=1..m} d(k), where d(k) is the decimal fraction 0.k (e.g. d(999)=0.999). 6

%I

%S 1,10,45,64,369,948,1155,9687,22998,126291,174997,1299997,4452157,

%T 6463650,29499996,69999996,888398929,4549999995,38445656295,

%U 454999999995,683977034682,699999999994,29499999999994,45426425047212,129999999999993,267746023852371,579369695158668

%N Integers of the form Sum_{k=1..m} d(k), where d(k) is the decimal fraction 0.k (e.g. d(999)=0.999).

%C These are the sums arising in A054464.

%H Robert Israel, <a href="/A275623/b275623.txt">Table of n, a(n) for n = 1..1793</a>

%F From _Robert Israel_, Aug 07 2016: (Start)

%F For d >=2, the m with d digits are the solutions of x^2 + x - 9*10^(d-1)*d - 10^(d-1) == 0 (mod 2*10^d) with 10^(d-1) <= x < 10^d.

%F The corresponding a(n) are m(m+1)10^(-d)/2 + (10^d-9d-1)/20. (End)

%p T:= (x, d) -> ((1/2)*x^2+(1/2)*x)*10^(-d)-(9/20)*d+(1/20)*10^d-1/20:

%p F:= proc(d) local x, S;

%p S:= map(t -> subs(t, x), [msolve(x^2 + x - 9*10^(d-1)*d - 10^(d-1), 2*10^d)]);

%p op(map(T, sort(select(t -> t >= 10^(d-1) and t < 10^d, S)), d))

%p end proc:

%p [1,op(map(F, [$2..30]))]; # _Robert Israel_, Aug 07 2016

%Y Cf. A054464, A275572.

%K nonn,base

%O 1,2

%A _N. J. A. Sloane_, Aug 07 2016

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Last modified November 28 12:00 EST 2021. Contains 349401 sequences. (Running on oeis4.)