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A328354
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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.
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8
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0, 1, 78, 5517, 386590, 27064101, 1894506678, 132615604717, 9283093290990, 649816537094901, 45487157643722278, 3184101035390113917, 222887072479614855390, 15602095073589188045701, 1092146655151356200377878, 76450265860595725286703117, 5351518610241706308890979790, 374606302716919480394120916501
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = (60*70^n-69*7^n+9) / 3726.
a(n) = 71*a(n-1) - 70*a(n-2) + 7^(n-1) for n > 1.
G.f.: x / (1 - 78*x + 567*x^2 - 490*x^3).
a(n) = 78*a(n-1) - 567*a(n-2) + 490*a(n-3) for n > 2.
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EXAMPLE
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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.
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MATHEMATICA
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LinearRecurrence[{78, -567, 490}, {0, 1, 78}, 20] (* Harvey P. Dale, Jan 30 2024 *)
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PROG
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(Python) [(60*70**n-69*7**n+9)//3726 for n in range(20)]
(PARI) concat(0, Vec(x / ((1 - x)*(1 - 7*x)*(1 - 70*x)) + O(x^17))) \\ Colin Barker, Dec 06 2019
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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