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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.

%I A328353 #22 Jun 01 2022 18:10:21
%S A328353 0,1,67,4063,244039,14643895,878643031,52718637847,3163118606743,
%T A328353 189787118420119,11387227117300375,683233627110581911,
%U A328353 40994017627070271127,2459641057626828406423,147578463457625377218199,8854707807457616670088855,531282468447457564427312791,31876948106847457250970656407
%N 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.
%C A328353 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.
%H A328353 Pierre-Alain Sallard, <a href="/A328353/b328353.txt">Table of n, a(n) for n = 0..50</a>
%H A328353 Pierre-Alain Sallard, <a href="/A328353/a328353.pdf">Integers sequences A328348 and A328350 to A328356</a>
%H A328353 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (67,-426,360).
%F A328353 a(n) = (50*60^n - 59*6^n + 9) / 2655.
%F A328353 a(n) = 61*a(n-1) - 60*a(n-2) + 6^(n-1) for n > 1.
%F A328353 G.f.: x / (1 - 67*x + 426*x^2 -360*x^3).
%F A328353 a(n) = 67*a(n-1) - 426*a(n-2) + 360*a(n-3) for n > 2.
%e A328353 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.
%t A328353 LinearRecurrence[{67,-426,360},{0,1,67},20] (* _Harvey P. Dale_, Feb 11 2022 *)
%o A328353 (Python) [(50*60**n-59*6**n+9)//2655 for n in range(20)]
%Y A328353 Cf. A328348, A328350, A328351, A328352, A328354, A328355, A328356.
%K A328353 nonn,base
%O A328353 0,3
%A A328353 _Pierre-Alain Sallard_, Nov 26 2019