A331673 -id:A331673 - cp's OEIS frontend

A130835 Sum of all numbers having n or fewer digits and having the sum of their digits equal to n.

1, 33, 1110, 38885, 1399986, 51333282, 1906666476, 71499999285, 2701111108410, 102631111100848, 3917722222183045, 150126888888738762, 5771538888888311735, 222499777777775552780, 8598259999999991401740, 332968856666666633369781, 12918171566666666537484951

Offset: 1

J. M. Bergot, Jul 18 2007

			Take n = 3. The numbers to be summed are 111, 3, 30, 300, 210, 201, 120, 102, 21 and 12, which add to 1110.

Alois P. Heinz, Table of n, a(n) for n = 1..626

Cf. A000042, A002275, A007953, A167403, A211072, A331672, A331673.

A007953 := proc(n) add(i,i=convert(n,base,10)) ; end: A130835 := proc(n) local a,i; a := 0 ; for i from 1 to 10^n-1 do if A007953(i) = n then a := a+i ; fi ; od ; RETURN(a) ; end: seq(A130835(n),n=1..4) ; # R. J. Mathar, Aug 01 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i=0, 0, add(b(n-j, i-1), j=0..min(n, 9))))
    end:
a:= n-> b(n, n)*(10^n-1)/9:
seq(a(n), n=1..20); # Alois P. Heinz, Nov 02 2009

a(n) = (10^n-1)/9 * [x^n] ((x^10-1)/(x-1))^n. - Alois P. Heinz, Feb 07 2012

a(n) = A000042(n) * A167403(n) = A002275(n) * A167403(n). - Alois P. Heinz, Aug 16 2018

a(4)-a(6) from R. J. Mathar, Aug 01 2007
a(7)-a(12) from Donovan Johnson, Jul 02 2009
More terms from Alois P. Heinz, Nov 02 2009

A331672 Sum of all base-n numbers with digit sum n and length at most n.

Original entry on oeis.org

3, 91, 2635, 94501, 4254936, 234572213, 15403880115, 1176838159861, 102631111100848, 10063085278250005, 1095923297151849530, 131253123286275198027, 17145216226230367266330, 2425892898650501790637545, 369599184391990522425455939, 60326656013944234430010524773

Offset: 2

Alois P. Heinz, Feb 22 2020

The cardinality of these numbers is given by A048775(n-1).

			a(2) = 3 = 11_2.
a(3) = 91 = 5 + 7 + 11 + 13 + 15 + 19 + 21 = 12_3 + 21_3 + 102_3 + 111_3 + 120_3 + 201_3 + 210_3.
a(10) = A130835(10) = 102631111100848.

Alois P. Heinz, Table of n, a(n) for n = 2..322

Cf. A007953, A023037, A048775, A130835, A331673.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
      `if`(i=0, 0, add((p->[p[1], p[2]*k+p[1]*d])(
         b(n-d, i-1, k)), d=0..min(n, k-1))))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=2..17);
# second Maple program:
a:= n-> (binomial(2*n-1, n)-n)*(n^n-1)/(n-1):
seq(a(n), n=2..17);

a(n) = A048775(n-1)*A023037(n) = (binomial(2*n-1,n)-n)*(n^n-1)/(n-1).

cp's OEIS Frontend

A130835 Sum of all numbers having n or fewer digits and having the sum of their digits equal to n.

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