A338271 a(n) is the number of compositions of n, b_1 + ... + b_t = n such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) is an integer.

1, 0, 0, 2, 0, 2, 0, 2, 2, 4, 2, 6, 2, 8, 4, 14, 6, 20, 8, 28, 14, 44, 20, 66, 30, 96, 46, 146, 70, 220, 102, 326, 154, 490, 232, 740, 346, 1102, 520, 1652, 782, 2484, 1166, 3716, 1750, 5568, 2628, 8358, 3936, 12518, 5900, 18760, 8848, 28138, 13256, 42170

Offset: 1

Author

Peter Kagey, Oct 19 2020

Keywords

• nonn

Comments

a(n) <= Sum_{k=1..floor(sqrt(n)/2)} A338286(floor((n-4*k^2)/2)) when n is even.

a(n) <= Sum_{k=1..floor((sqrt(n) - 1)/2)} A338286(floor((n-4*k^2-4*k-1)/2)) when n is odd and greater than 1.

Examples

			(Let s(k) = sqrt(k) for brevity.)
For n = 14, the a(14) = 8 valid compositions are:
14 = 2+2+2+2+2+3+1 and 2 = s(2+s(2+s(2+s(2+s(2+s(3+s(1)))))))
14 = 1+7+2+3+1     and 2 = s(1+s(7+s(2+s(3+s(1)))))
14 = 2+1+7+3+1     and 2 = s(2+s(1+s(7+s(3+s(1)))))
14 = 2+2+1+8+1     and 2 = s(2+s(2+s(1+s(8+s(1)))))
14 = 2+2+2+2+2+4   and 2 = s(2+s(2+s(2+s(2+s(2+s(4))))))
14 = 1+7+2+4       and 2 = s(1+s(7+s(2+s(4))))
14 = 2+1+7+4       and 2 = s(2+s(1+s(7+s(4))))
14 = 2+2+1+9       and 2 = s(2+s(2+s(1+s(9))))
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..500
• Code Golf Stack Exchange, The square root of the square root of the square root of the...

Crossrefs

Cf. A000196, A338268.

Cf. A098504, A101391, A143862, A238351, A238686, A325676. A335942.

Formula

a(n) = Sum_{i=k..A000196(n)} A338268(n,k).