A243312 - OEIS

%I #22 Feb 13 2021 10:21:33

%S 1,4,31,305,3345,39505,487935,6245118

%N The total T(d,n) of expressions with n instances of a digit d between 6 and 9 (and using the four arithmetic operators) which have a defined value when evaluated.

%C Bounded above by A151403 (which is equal to the *total* number of expressions, not just the ones that evaluate to a valid numeric value).

%C An expression has an undefined value when it contains a "division by zero" operation.

%C For 0 <= d <= 5 and d' > 5, it is possible that T(d,n) < T(d',n). See link below.

%C T(6,10) = 1097128463 < T(7,10) = 1097153419 < T(8,10) = 1097155971 < T(9,10) = 1097157195. - _Hiroaki Yamanouchi_, Oct 01 2014

%H David Lyness, <a href="https://davidlyness.com/post/experimenting-bracketing-binary-operators">Experimenting with bracketing and binary operators</a>

%e For n = 2 and digit d != 0, there are four possible expressions:

%e (d + d)

%e (d - d)

%e (d * d)

%e (d / d)

%e None of these include a "division by zero" operation, and so all four of the above expressions can be considered valid. Therefore, T(d, 2) = 4 for d > 0.

%o (Python)

%o # coding=utf-8

%o .

%o import itertools

%o .

%o .

%o def all_expressions(generic_expression, operation_combinations):

%o ...."""

%o ....Merges a source expression and combinations of binary operators to generate a list of all possible expressions.

%o ....@param ((str,)) operation_combinations: all combinations of binary operators to consider

%o ....@param str generic_expression: source expression with placeholder binary operators

%o ....@rtype: (str,)

%o ...."""

%o ....expression_combinations = []

%o ....for combination in operation_combinations:

%o ........expression_combinations.append(generic_expression.format(*combination))

%o ....return expression_combinations

%o .

%o .

%o def all_bracketings(expr):

%o ...."""

%o ....Generates all possible permutations of parentheses for an expression.

%o ....@param str expr: the non-bracketed source expression

%o ....@rtype: str

%o ...."""

%o ....if len(expr) == 1:

%o ........yield expr

%o ....else:

%o ........for i in range(1, len(expr), 2):

%o ............for left_expr in all_bracketings(expr[:i]):

%o ................for right_expr in all_bracketings(expr[i + 1:]):

%o ....................yield "({}{}{})".format(left_expr, expr[i], right_expr)

%o .

%o .

%o def num_valid_expressions(num_digits):

%o ...."""Perform all calculations with the given operations and in the range of digits specified.

%o ....@param int num_digits: the number of digits in the expression

%o ....@rtype: int

%o ...."""

%o ....operations = ["+", "-", "*", "/"]

%o ....digit = 9

%o ....operation_iterable = itertools.product(*[operations] * (num_digits - 1))

%o ....operation_combinations = []

%o ....valid_expression_count = 0

%o ....template = " ".join(str(digit) * num_digits)

%o ....for operation in operation_iterable:

%o ........operation_combinations.append(operation)

%o ....for bracketed_expression in all_bracketings(template):

%o ........for expression in all_expressions(bracketed_expression.replace(" ", "{}"), operation_combinations):

%o ............try:

%o ................eval(expression)

%o ................valid_expression_count += 1

%o ............except ZeroDivisionError:

%o ................pass

%o ....return valid_expression_count

%o .

%o .

%o if __name__ == "__main__":

%o ....min_num_digits = 1

%o ....max_num_digits = 6

%o ....operation_set = ["+", "-", "*", "/"]

%o ....for num in range(min_num_digits, max_num_digits + 1):

%o ........print(num_valid_expressions(num))

%K nonn,base,uned,obsc,more

%O 1,2

%A _David Lyness_, Jun 03 2014