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For the rules of the Jumping Frogs game, see A377232.

Given the result that any block of consecutive single frogs is a winning position, the "interesting" positions are those that have an empty place, or gap, in them. The value a(n) measures "how difficult" the n-th such position is, as follows: Starting from a nonnegative integer n, interpret its binary representation as a Jumping Frogs position consisting of empty and/or singly-filled places. Add one more empty place (one more 0) at the end, to ensure that there is at least one gap. The value a(n) is then the minimum positive number of additional singly-filled places that must be added thereafter to create a winning position, or -1 if no winning position can be so constructed.

Equivalently, a(n) is the least k such that n*2^{k+1} + 2^k - 1 is a term of A377232, should such a k exist, and -1 otherwise.

Gordon Hamilton conjectures in his reference below that a(n) is positive for all n.

For the rules of the Jumping Frogs game, see A377232.

Counts the number of distinct winning positions of the game with exactly n frogs, each in a different place. Positions are considered the same if one can be obtained from the other by reversing it left-to-right.

Equivalently, a(n) is the number of unordered pairs {k, A030101(k)} with k odd such that the binary weight A000120(k) = n, and k occurs in A377232.

Since a stack of k frogs moves exactly k spaces in the Jumping Frogs game, the sum of the lengths of all moves starting from a position with n single frogs is at most A000217(n-1), the (n-1)st triangular number. Hence if it is a winning position, the length of the position (excluding unoccupied places to the left of the first occupied one, and unoccupied places to the right of the last occupied one) is at most one more than this triangular number. Therefore a(n) is finite for all n.

For the rules of the Jumping Frogs game, see A377232.

Enumerate the primes in order, p_1 = 2, p_2 = 3, etc. Factor any natural number k > 1 as p_1^{x_1}p_2^{x_2}...p_i^{x_i}, where i is as small as possible and each x_j is nonnegative. Then when k is even and x_1, x_2, ..., x_i is a winning position for Jumping Frogs, k occurs as a term. We consider only even numbers to keep the positions distinct; leading zeros can never be used or affect the outcome of Jumping Frogs.

An even number k is a term if and only if A137502(k) is a term. - Pontus von Brömssen, Oct 24 2024

For the rules of the Jumping Frogs game, see A377232.

A winning position and its reversal are both counted.

a(n) is the number of terms x of A377232 in the interval 2^n <= x < 2^(n+1).