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A328355
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Let S be any integer in the range 36 <= S <= 44. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and uses eight distinct nonzero digits d1,d2,d3,d4,d5,d6,d7,d8 such that d1+d2+d3+d4+d5+d6+d7+d8=S.
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8
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0, 1, 89, 7193, 576025, 46086681, 3686971929, 294958053913, 23596646709785, 1887731755956761, 151018540629932569, 12081483251621739033, 966518660139556190745, 77321492811243031804441, 6185719424900070836714009, 494857553992010693275990553, 39588604319360895672790202905
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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 expansion uses exactly eight distinct, nonzero digits: see the attached pdf documents.
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LINKS
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FORMULA
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a(n) = (70*80^n - 79*8^n + 9) / 4977.
a(n) = 81 a(n-1) - 80 a(n-2) + 8^(n-1) for n > 1.
G.f.: x / (1 - 89*x + 728*x^2 - 640*x^3).
a(n) = 89*a(n-1) - 728*a(n-2) + 640*a(n-3) for n > 2.
E.g.f.: (9*exp(x) - 79*exp(8*x) + 70*exp(80*x))/4977. - Stefano Spezia, Dec 11 2019
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EXAMPLE
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For n=2, the sum of all positive integers whose decimal notation is made of any digit different from 0 and, let's say, 9 with at most n=2 such digits, i.e., the sum 1+2+3+4+5+6+7+8+11+12+13+14+15+16+17+18+21+...+28+31+...+38+41+...+48+51+...+58+61+...+68+71+...+78+81+...+88, is equal to a(2)*(1+2+3+4+5+6+7+8) = 89*36 = 3204.
Similarly, and always with n=2, the sum of all positive integers whose decimal notation is made of any digit different from 0 and, let's say, 8, i.e., the sum 1+2+3+4+5+6+7+9+11+..+17+19+21+...+27+29+31+...+37+39+41+...+47+49+51+...+57+59+61+...+67+69+71+...+77+79+91+...+97+99 is equal to a(2)*(1+2+3+4+5+6+7+9) = 89*37 = 3293.
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MATHEMATICA
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CoefficientList[Series[x/(1 - 89 x + 728 x^2 - 640 x^3), {x, 0, 16}], x] (* Michael De Vlieger, Dec 10 2019 *)
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PROG
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(Python)[(70*80**n-79*8**n+9)//4977 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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