A279766 -id:A279766 - cp's OEIS frontend

A358854 Number of even digits necessary to write all the numbers from 0 up to n.

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 66, 66, 67, 67, 68, 68

Offset: 0

Bernard Schott, Dec 03 2022

Inspired by problem 1 of British Mathematical Olympiad, round 1, in 2016/2017 (link).

Alois P. Heinz, Table of n, a(n) for n = 0..20000
United Kingdom Mathematics Trust, 2016/17 British Mathematical Olympiad Round 1, Problem 1
Index to sequences related to Olympiads.

Cf. A117804, A196563, A279766.

a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+
      nops(select(x-> x::even, convert(n, base, 10))))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 19 2023

Accumulate[Table[Count[IntegerDigits[n], ?EvenQ], {n, 0, 100}]] (* _Amiram Eldar, Dec 03 2022 *)

from itertools import accumulate, count, islice
def A196563(n): return sum(1 for d in str(n) if d in "02468")
def agen(): yield from accumulate(A196563(n) for n in count(0))
print(list(islice(agen(), 76))) # Michael S. Branicky, Dec 03 2022

a(n) = A117804(n+1) - A279766(n) (number of total digits - number of odd digits).

a(n) = Sum_{k=0..n} A196563(k).

A359271 Number of odd digits necessary to write all nonnegative n-digit integers.

Original entry on oeis.org

5, 95, 1400, 18500, 230000, 2750000, 32000000, 365000000, 4100000000, 45500000000, 500000000000, 5450000000000, 59000000000000, 635000000000000, 6800000000000000, 72500000000000000, 770000000000000000, 8150000000000000000

Offset: 1

Bernard Schott, Dec 23 2022

			To write the integers from 10 up to 99, each of the digits 1, 3, 5, 7 and 9, must be used 19 times, hence a(2) = 19*5 = 95.

Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A113119, A196564, A279766, A358439.

Maple
```
seq(5 * (9*n+1) * 10^(n-2), n=1..18);
```

a[n_] := 5*(9*n + 1)*10^(n - 2); Array[a, 20] (* Amiram Eldar, Dec 23 2022 *)

a(n) = 5 * (9*n+1) * 10^(n-2).
a(n) = A279766(10^n-1) - A279766(10^(n-1)-1).
a(n) = A113119(n) - A358439(n).
From Stefano Spezia, Dec 24 2022: (Start)
O.g.f.: 5*x*(1 - x)/(1 - 10*x)^2.
E.g.f.: (exp(10*x)*(1 + 90*x) - 1)/20. (End)

cp's OEIS Frontend

A358854 Number of even digits necessary to write all the numbers from 0 up to n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula