A078427 Sum of all the decimal digits of numbers from 1 to 10^n.

46, 901, 13501, 180001, 2250001, 27000001, 315000001, 3600000001, 40500000001, 450000000001, 4950000000001, 54000000000001, 585000000000001, 6300000000000001, 67500000000000001, 720000000000000001, 7650000000000000001, 81000000000000000001, 855000000000000000001

Offset

1,1

References

Edward J. Barbeau et al., Five Hundred Mathematical Challenges, Problem 284. pp. 25; 142-143, MAA Washington DC, 1995.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Formula

a(n) = 45*n*10^(n-1) + 1.

a(n) = 45*A053541(n) + 1. - Lekraj Beedassy, Sep 16 2006

From Colin Barker, May 23 2014: (Start)

a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3).

G.f.: -x*(100*x^2-65*x+46)/((x-1)*(10*x-1)^2). (End)

E.g.f.: exp(x)*(45*x*exp(9*x) + 1) - 1. - Elmo R. Oliveira, Nov 20 2025

a(n) = A034967(n) + 1. - Alois P. Heinz, Nov 20 2025

Example

a(2)=901 because sum of all the digits of numbers from 1 to 10^2 is 901.

Mathematica

LinearRecurrence[{21, -120, 100}, {46, 901, 13501}, 20] (* Harvey P. Dale, Nov 24 2016 *)

Prog

(Magma) [(45*n)*10^(n-1)+1: n in [1..30]]; // Vincenzo Librandi, Jun 06 2011
(PARI) Vec(-x*(100*x^2-65*x+46)/((x-1)*(10*x-1)^2) + O(x^100)) \\ Colin Barker, May 23 2014

Crossrefs

Cf. A034967, A053541, A072290.

Sequence in context: A113922 A160067 A156842 * A002138 A205495 A281655

Adjacent sequences: A078424 A078425 A078426 * A078428 A078429 A078430

Keyword

nonn,base,easy

Author

Shyam Sunder Gupta, Dec 29 2002

Extensions

Edited by Charles R Greathouse IV, Aug 02 2010

More terms from Colin Barker, May 23 2014

Status

approved