Euler's partition theorem and the combinatorics of -sequences

l = 3
The l-Euler theorem [BME2]: The number of partitions of an
integer N into parts from the set
{0 + 1, 1 + 3, 3 + 8, . . .} = {1, 4, 11, 29, . . .}
is the same as the number of partitions of N in which the ratio of
consecutive parts is greater than
c 3 = 3 + √ 3 2 − 4
2
= (3 + √ 5)/2