A192107 Sum of all the n-digit numbers whose digits are all odd.

25, 1375, 69375, 3471875, 173609375, 8680546875, 434027734375, 21701388671875, 1085069443359375, 54253472216796875, 2712673611083984375, 135633680555419921875, 6781684027777099609375, 339084201388885498046875, 16954210069444427490234375

Offset

1,1

Comments

The idea for this sequence comes from question 4 of the Final Round of the Finnish High School Mathematics Contest in 1997 (see link IMO Compendium and Crux reference) where the question was asked regarding only 4-digit numbers.

A192370 is the similar sequence when all the digits are even: 2, 4, 6, 8.

A220094 is the similar sequence with the digits belonging to {1, 2, 3, 4, 5, 6, 7, 8, 9}.

References

Finnish High School Mathematics Contest, Final Round, 1997, problem 4. [Crux Mathematicorum, v22 n3, Apr. 2002, p. 143]

Links

Table of n, a(n) for n=1..15.

The IMO compendium, Problem 4, Finnish High School Mathematics Contest 1997.

Index entries for linear recurrences with constant coefficients, signature (55,-250).

Index to sequences related to Olympiads.

Formula

a(n) = ((10^n-1) * 5^(n+1))/9 = 5^(n+1) * R_n with R_n is the repunit with n times the digit 1.

From Colin Barker, Jan 04 2013: (Start)

a(n) = 55*a(n-1) - 250*a(n-2).

G.f.: 25*x/((5*x-1)*(50*x-1)). (End)

Example

a(1) = 1 + 3 + 5 + 7 + 9 = 25.
a(2) = 11 + 13 + ... + 19 + 31 + ... + 79 + 91 + ... + 99 = 1375.

Maple

A:=seq((10^n-1)*5^(n+1)/9, n=1..20);

Mathematica

Table[((10^n - 1)*5^(n + 1))/9, {n, 20}] (* T. D. Noe, Dec 31 2012 *)
LinearRecurrence[{55, -250}, {25, 1375}, 20] (* Harvey P. Dale, Oct 11 2018 *)

Prog

(PARI) a(n) = (10^n-1) * 5^(n+1)/9 \\ Charles R Greathouse IV, Jul 06 2017

Crossrefs

Cf. A220094, A192370, A014261.

Sequence in context: A046910 A278851 A243943 * A206465 A364244 A138246

Adjacent sequences: A192104 A192105 A192106 * A192108 A192109 A192110

Keyword

nonn,base,easy

Author

Bernard Schott, Dec 30 2012

Status

approved