A328353 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses six distinct nonzero digits d1,d2,d3,d4,d5,d6 such that d1+d2+d3+d4+d5+d6=S.

0, 1, 67, 4063, 244039, 14643895, 878643031, 52718637847, 3163118606743, 189787118420119, 11387227117300375, 683233627110581911, 40994017627070271127, 2459641057626828406423, 147578463457625377218199, 8854707807457616670088855, 531282468447457564427312791, 31876948106847457250970656407

Offset: 0

Author

Pierre-Alain Sallard, Nov 26 2019

Keywords

• nonn
• base

Comments

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

Examples

			For n=2, the sum of all positive integers whose decimal notation is only made of, let's say, the 4,5,6,7,8,9 digits with at most n=2 such digits, i.e. the sum 4+5+6+7+8+9+44+45+46+47+48+49+54+55+56+57+58+59+64+65+66+67+68+69+74+75+76+77+78+79+84+85+86+87+88+89+94+95+96+97+98+99 is equal to a(2)*(4+5+6+7+8+9) = 67*39 = 2613.
		

Links

• Pierre-Alain Sallard, Table of n, a(n) for n = 0..50
• Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356
• Index entries for linear recurrences with constant coefficients, signature (67,-426,360).

Crossrefs

Cf. A328348, A328350, A328351, A328352, A328354, A328355, A328356.

Programs

Mathematica

LinearRecurrence[{67,-426,360},{0,1,67},20] (* Harvey P. Dale, Feb 11 2022 *)

Python

[(50*60**n-59*6**n+9)//2655 for n in range(20)]

Formula

a(n) = (50*60^n - 59*6^n + 9) / 2655.

a(n) = 61*a(n-1) - 60*a(n-2) + 6^(n-1) for n > 1.

G.f.: x / (1 - 67*x + 426*x^2 -360*x^3).

a(n) = 67*a(n-1) - 426*a(n-2) + 360*a(n-3) for n > 2.