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A331529
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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 (1st version).
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4
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0, 0, 1, 1, 2, 5, 7, 12, 19, 33, 59, 99, 170, 290, 496, 854, 1463, 2506, 4292, 7351, 12601, 21596, 37005, 63405, 108637, 186154, 318989, 546600, 936606, 1604874, 2749973, 4712146, 8074374, 13835600, 23707533, 40623267, 69608738, 119275933, 204381606, 350211711, 600094277
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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 A006942, where a 7 is depicted by 3 segments.
Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = f(6) = 3 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 first formula).
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,3,3,1).
Index entries for sequences related to compositions
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FORMULA
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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 + 3*x^5 + 3*x^6 + x^7.
From Colin Barker, Jan 20 2020: (Start)
G.f.: x^2*(1 - x)*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4) / (1 - x^2 - x^3 - x^4 - 3*x^5 - 3*x^6 - x^7).
a(n) = a(n-2) + a(n-3) + a(n-4) + 3*a(n-5) + 3*a(n-6) + a(n-7) for n>13.
(End)
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EXAMPLE
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a(5) = 5 since 2, 3, 5, 17 and 71 are displayed by 5 segments.
__ __ __ __ __
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(2) (3) (5) (17) (71)
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MATHEMATICA
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P[x_]:=x^2+x^3+x^4+3x^5+3x^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, 41, 0]
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PROG
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(PARI) concat([0, 0], Vec(x^2*(1 - x)*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4) / (1 - x^2 - x^3 - x^4 - 3*x^5 - 3*x^6 - x^7) + O(x^41))) \\ Colin Barker, Jan 20 2020
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CROSSREFS
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Cf. A002426, A004526, A006942, A216261, A331530 (2nd version).
Sequence in context: A042343 A042691 A112732 * A042065 A041793 A333068
Adjacent sequences: A331526 A331527 A331528 * A331530 A331531 A331532
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KEYWORD
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base,nonn,easy
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AUTHOR
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Stefano Spezia, Jan 19 2020
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STATUS
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approved
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