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).

1, 10, 45, 64, 369, 948, 1155, 9687, 22998, 126291, 174997, 1299997, 4452157, 6463650, 29499996, 69999996, 888398929, 4549999995, 38445656295, 454999999995, 683977034682, 699999999994, 29499999999994, 45426425047212, 129999999999993, 267746023852371, 579369695158668

Offset

1,2

Comments

These are the sums arising in A054464.

Links

Robert Israel, Table of n, a(n) for n = 1..1793

Formula

From Robert Israel, Aug 07 2016: (Start)

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.

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

Maple

T:= (x, d) -> ((1/2)*x^2+(1/2)*x)*10^(-d)-(9/20)*d+(1/20)*10^d-1/20:
F:= proc(d) local x, S;
  S:= map(t -> subs(t, x), [msolve(x^2 + x - 9*10^(d-1)*d - 10^(d-1), 2*10^d)]);
  op(map(T, sort(select(t -> t >= 10^(d-1) and t < 10^d, S)), d))
end proc:
[1, op(map(F, [$2..30]))]; # Robert Israel, Aug 07 2016

Crossrefs

Cf. A054464, A275572.

Sequence in context: A299530 A238982 A292611 * A245871 A199350 A199516

Adjacent sequences: A275620 A275621 A275622 * A275624 A275625 A275626

Keyword

nonn,base

Author

N. J. A. Sloane, Aug 07 2016

Status

approved