login
A342602
Number of solutions to 1 +-* 2 +-* 3 +-* ... +-* n = 0.
2
0, 0, 1, 1, 1, 4, 6, 14, 29, 63, 129, 300, 756, 1677, 4134, 9525, 22841, 57175, 141819, 354992, 882420, 2218078, 5588989, 14173217, 35918542
OFFSET
1,6
COMMENTS
Normal operator precedence is followed, so multiplication is performed before addition or subtraction. Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 3.
The author thanks Ursula Ponting for useful discussions.
EXAMPLE
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(5) = 1 as 1 * 2 - 3 - 4 + 5 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(6) = 4. The solutions, all of which use multiplication, are
1 + 2 * 3 + 4 - 5 - 6 = 0,
1 - 2 + 3 * 4 - 5 - 6 = 0,
1 - 2 * 3 + 4 - 5 + 6 = 0,
1 * 2 + 3 - 4 + 5 - 6 = 0.
a(10) = 63. An example solution is
1 - 2 * 3 * 4 - 5 - 6 - 7 * 8 + 9 * 10 = 0.
a(20) = 354992. 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 includes thirteen multiplications.
MATHEMATICA
Table[Length@Select[Tuples[{"+", "-", "*"}, k-1], ToExpression[""<>Riffle[ToString/@Range@k, #]]==0&], {k, 11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)
PROG
(Python)
from itertools import product
def a(n):
nn = [str(i) for i in range(1, n+1)]
return sum(eval("".join([*sum(zip(nn, ops+("", )), ())])) == 0 for ops in product("+-*", repeat=n-1))
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Apr 02 2021
CROSSREFS
Cf. A342804 (using +-*/), A342995 (using +-/), A058377 (using +-), A063865, A000217, A025591, A161943.
Sequence in context: A303041 A103419 A005202 * A106526 A319766 A373567
KEYWORD
nonn,more
AUTHOR
Scott R. Shannon, Mar 27 2021
STATUS
approved