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A343314
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a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A063720).
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5
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0, 0, 1, 1, 2, 7, 5, 16, 19, 39, 77, 103, 226, 334, 636, 1106, 1827, 3386, 5568, 10059, 17281, 29890, 52771, 90283, 159191, 274976, 479035, 835476, 1447278, 2528496, 4386143, 7640592, 13293308, 23106132, 40245277, 69946521, 121762316, 211791205, 368418674, 641125867
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OFFSET
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0,5
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COMMENTS
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The nonnegative integers are displayed as in A063720.
Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = 5 and f(s) = 1 elsewhere, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see third formula).
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LINKS
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FORMULA
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G.f.: x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + 5*x^3 + x^4 + x^5)/(1 - x^2 - x^3 - x^4 - 5*x^5 - x^6 - x^7).
a(n) = a(n-2) + a(n-3) + a(n-4) + 5*a(n-5) + a(n-6) + a(n-7) for n > 13.
a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = x^2 + x^3 + x^4 + 5*x^5 + x^6 + x^7.
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EXAMPLE
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a(6) = 5 since 0, 14, 41, 77 and 111 are displayed by 6 segments.
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(0) (14) (41) (77) (111)
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MATHEMATICA
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P[x_]:=x^2+x^3+x^4+5x^5+x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; a[n_]:=b[n]-b[n-6]; Array[a, 40, 0]
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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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