A338268 -id:A338268 - cp's OEIS frontend

A338286 a(n) = A338268(k^2 + 2*n, k) for sufficiently large k.

2, 2, 2, 4, 6, 8, 14, 20, 28, 44, 68, 98, 148, 224, 332, 500, 754, 1122, 1682, 2530, 3786, 5674, 8514, 12752, 19112, 28668, 42962, 64386, 96538, 144702, 216888, 325150, 487386, 730548, 1095140, 1641616, 2460710, 3688646, 5529280, 8288266, 12424144, 18623788

Offset: 0

Peter Kagey, Oct 20 2020

nonn

Here, "sufficiently large k" means k >= 1 + sqrt((1 + sqrt(1 + 8 * n))/2).

Peter Kagey, Table of n, a(n) for n = 0..250

Cf. A338268.

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.

Original entry on oeis.org

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

Peter Kagey, Oct 19 2020

nonn

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.

			(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))))

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...

Cf. A000196, A338268.

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

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

cp's OEIS Frontend

A338286 a(n) = A338268(k^2 + 2*n, k) for sufficiently large k.

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