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A328354 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses seven distinct nonzero digits d1,d2,d3,d4,d5,d6,d7 such that d1+d2+d3+d4+d5+d6+d7=S.

%I A328354 #26 Jan 30 2024 13:47:00
%S A328354 0,1,78,5517,386590,27064101,1894506678,132615604717,9283093290990,
%T A328354 649816537094901,45487157643722278,3184101035390113917,
%U A328354 222887072479614855390,15602095073589188045701,1092146655151356200377878,76450265860595725286703117,5351518610241706308890979790,374606302716919480394120916501
%N A328354 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses seven distinct nonzero digits d1,d2,d3,d4,d5,d6,d7 such that d1+d2+d3+d4+d5+d6+d7=S.
%C A328354 This sequence is the building block for the calculation of the sums of positive integers whose decimal notation uses seven distinct, nonzero digits: see the attached pdf document.
%H A328354 Pierre-Alain Sallard, <a href="/A328354/b328354.txt">Table of n, a(n) for n = 0..50</a>
%H A328354 Pierre-Alain Sallard, <a href="/A328354/a328354.pdf">Integers sequences A328348 and A328350 to A328356</a>
%H A328354 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (78,-567,490).
%F A328354 a(n) = (60*70^n-69*7^n+9) / 3726.
%F A328354 a(n) = 71*a(n-1) - 70*a(n-2) + 7^(n-1) for n > 1.
%F A328354 G.f.: x / (1 - 78*x + 567*x^2 - 490*x^3).
%F A328354 a(n) = 78*a(n-1) - 567*a(n-2) + 490*a(n-3) for n > 2.
%e A328354 For n=2, the sum of all positive integers whose decimal notation is made of, let's say, the 3,4,5,6,7,8 and 9 digits with at most n=2 such digits, i.e., the sum 3+4+5+6+7+8+9+33+34+35+36+37+38+39+43+44+45+46+47+48+49+53+54+55+56+57+58+59+63+64+65+66+67+68+69+73+74+75+76+78+79+83+84+85+86+87+88+89+93+94+95+96+97+98+99, is equal to a(2)*(3+4+5+6+7+8+9) = 78*42 = 3276.
%t A328354 LinearRecurrence[{78,-567,490},{0,1,78},20] (* _Harvey P. Dale_, Jan 30 2024 *)
%o A328354 (Python) [(60*70**n-69*7**n+9)//3726 for n in range(20)]
%o A328354 (PARI) concat(0, Vec(x / ((1 - x)*(1 - 7*x)*(1 - 70*x)) + O(x^17))) \\ _Colin Barker_, Dec 06 2019
%Y A328354 Cf. A328348, A328350, A328351, A328352, A328353, A328355, A328356.
%K A328354 nonn,base,easy
%O A328354 0,3
%A A328354 _Pierre-Alain Sallard_, Nov 26 2019