A342602 Number of solutions to 1 +-* 2 +-* 3 +-* ... +-* n = 0.

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

Author

Scott R. Shannon, Mar 27 2021

Keywords

• nonn
• more

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.

Examples

			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.
		

Crossrefs

Cf. A342804 (using +-*/), A342995 (using +-/), A058377 (using +-), A063865, A000217, A025591, A161943.

Programs

Mathematica

Table[Length@Select[Tuples[{"+","-","*"},k-1],ToExpression[""<>Riffle[ToString/@Range@k,#]]==0&],{k,11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

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