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A342995
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Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.
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2
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0, 0, 1, 1, 0, 1, 4, 8, 0, 3, 37, 80, 6, 17, 461, 868, 190, 364, 5570, 11342, 3993, 7307, 78644
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OFFSET
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1,7
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COMMENTS
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Normal operator precedence is followed, so division is performed before addition or subtraction. Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 10.
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LINKS
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EXAMPLE
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a(3) = 1 as 1 + 2 - 3 = 0 is the only solution.
a(4) = 1 as 1 - 2 - 3 + 4 = 0 is the only solution.
a(6) = 1 as 1 - 2 / 3 / 4 - 5 / 6 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(8) = 8. Seven of the solutions involve just addition and subtraction, matching those in A058377, but one additional solution exists using division:
1 / 2 / 3 / 4 + 5 / 6 - 7 / 8 = 0.
a(10) = 3. All three solutions require division:
1 + 2 / 3 / 4 + 5 / 6 + 7 - 8 + 9 - 10 = 0,
1 - 2 / 3 / 4 - 5 / 6 + 7 - 8 - 9 + 10 = 0,
1 - 2 / 3 / 4 - 5 / 6 - 7 + 8 + 9 - 10 = 0.
a(15) = 461. Of these, 361 use only addition and subtraction, the other 100 also require division. One example of the latter is
1 / 2 / 3 / 4 - 5 - 6 - 7 / 8 + 9 / 10 + 11 + 12 - 13 + 14 / 15 = 0.
a(20) = 11342. An example solution is
1 / 2 / 3 - 4 / 5 / 6 + 7 / 8 / 9 + 10 + 11 / 12 - 13 + 14 / 15 / 16
+ 17 / 18 + 19 / 20 = 0
which sums seven fractions that include eleven divisions.
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MATHEMATICA
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Table[Length@Select[Tuples[{"+", "-", "/"}, k-1], ToExpression[""<>Riffle[ToString/@Range@k, #]]==0&], {k, 11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)
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PROG
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(Python)
from itertools import product
from fractions import Fraction
def a(n):
nn = ["Fraction("+str(i)+", 1)" for i in range(1, n+1)]
return sum(eval("".join([*sum(zip(nn, ops+("", )), ())])) == 0 for ops in product("+-/", repeat=n-1))
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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