A328350 Let S be any integer in the range 6 <= S <= 24. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and contains at most three distinct nonzero digits d1, d2, d3 such that d1+d2+d3 = S.

0, 1, 34, 1033, 31030, 931021, 27930994, 837930913, 25137930670, 754137929941, 22624137927754, 678724137921193, 20361724137901510, 610851724137842461, 18325551724137665314, 549766551724137133873, 16492996551724135539550, 494789896551724130756581, 14843696896551724116407674, 445310906896551724073360953

Offset

0,3

Comments

This sequence is the building block for the calculation of the sums of positive integers whose decimal expansion contains only three distinct, nonzero digits: see attached pdf document.

Links

Table of n, a(n) for n=0..19.

Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356

Index entries for linear recurrences with constant coefficients, signature (34,-123,90).

Formula

a(n) = (20*30^n - 29*3^n + 9)/522.

From Stefano Spezia, Oct 17 2019: (Start)

O.g.f.: x/(1 - 34*x + 123*x^2 - 90*x^3).

E.g.f.: (1/522)*(9*exp(x) - 29*exp(3*x) + 20*exp(30*x)).

a(n) = 34*a(n-1) - 123*a(n-2) + 90*a(n-3) for n > 2. (End)

a(n) = 31*a(n-1) - 30*a(n-2) + 3^n for n > 1. - Pierre-Alain Sallard, Dec 15 2019

Example

For n=2, the sum of all positive integers whose decimal expansions consist of at most n=2 of the digits 5, 6 or 7, i.e., the sum 5+6+7+55+56+57+65+66+67+75+76+77, is equal to a(2)*(5+6+7) = 612.
The formula is valid for any other choice of three distinct digits. Another example: again with n=2, but let's say with the digits 1, 2 and 3, the sum 1+2+3+11+12+13+21+22+23+31+32+33 is equal to a(2)*(1+2+3) = 204.

Mathematica

Array[(20*30^# - 29*3^# + 9)/522 &, 20, 0] (* or *)
LinearRecurrence[{34, -123, 90}, {0, 1, 34}, 20] (* Paolo Xausa, Apr 29 2024 *)

Prog

(Python) [(20*30**n-29*3**n+9)//522 for n in range(20)]

Crossrefs

Cf. A328348, A328351, A328352, A328353, A328354, A328355, A328356.

Sequence in context: A098607 A075292 A297456 * A365993 A275294 A296724

Adjacent sequences: A328347 A328348 A328349 * A328351 A328352 A328353

Keyword

nonn,base,easy

Author

Pierre-Alain Sallard, Oct 13 2019

Extensions

Needs editing. - N. J. A. Sloane, Dec 12 2019

Status

approved