A334635 Number of different values of x_1*x_2*...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.

1, 2, 4, 8, 15, 28, 51, 92, 164, 286, 486, 808, 1322, 2142, 3456, 5571, 8975, 14427, 23094, 36766, 58201, 91675, 143841, 225045, 351321, 547393, 851160, 1320339, 2042195, 3147614, 4831237, 7380048, 11213838, 16942556, 25447992, 38000880

Offset: 1

Author

Jack Zhang, Sep 10 2020

Keywords

• nonn

Comments

a(n) is also the number of different possible products of nesting levels in n pairs of parentheses.

Examples

			n=5: possible values are 1*1*1*1*1, 1*1*1*1*2, 1*1*1*2*2, 1*1*1*2*3, 1*1*2*2*2, 1*1*2*2*3, 1*1*2*3*3, 1*1*2*3*4, 1*2*2*2*2, 1*2*2*2*3, 1*2*2*3*3, 1*2*2*3*4, 1*2*3*3*3, 1*2*3*3*4, 1*2*3*4*4, 1*2*3*4*5, but since 1*1*2*3*4=1*2*2*2*3, there are only 15 different values.
		

Programs

Python

k=[{(1,1)}]
for i in range(20):
    k.append(set([(i[0]*i[1],i[1]) for i in k[-1]])|set([(i[0]*(i[1]+1),i[1]+1) for i in k[-1]]))
[len(set(j[0] for j in i)) for i in k]

Extensions

a(31)-a(32) from David A. Corneth, Sep 12 2020

a(33)-a(36) from Bert Dobbelaere, Oct 19 2020