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A328353
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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.
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8
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0, 1, 67, 4063, 244039, 14643895, 878643031, 52718637847, 3163118606743, 189787118420119, 11387227117300375, 683233627110581911, 40994017627070271127, 2459641057626828406423, 147578463457625377218199, 8854707807457616670088855, 531282468447457564427312791, 31876948106847457250970656407
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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 only uses six distinct, nonzero digits: see the attached pdf document.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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LinearRecurrence[{67, -426, 360}, {0, 1, 67}, 20] (* Harvey P. Dale, Feb 11 2022 *)
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PROG
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(Python) [(50*60**n-59*6**n+9)//2655 for n in range(20)]
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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