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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.

%I A338271 #16 Nov 02 2020 02:10:22
%S A338271 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,
%T A338271 70,220,102,326,154,490,232,740,346,1102,520,1652,782,2484,1166,3716,
%U A338271 1750,5568,2628,8358,3936,12518,5900,18760,8848,28138,13256,42170
%N 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.
%C A338271 a(n) <= Sum_{k=1..floor(sqrt(n)/2)} A338286(floor((n-4*k^2)/2)) when n is even.
%C A338271 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.
%H A338271 Peter Kagey, <a href="/A338271/b338271.txt">Table of n, a(n) for n = 1..500</a>
%H A338271 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/213848/53884">The square root of the square root of the square root of the...</a>
%F A338271 a(n) = Sum_{i=k..A000196(n)} A338268(n,k).
%e A338271 (Let s(k) = sqrt(k) for brevity.)
%e A338271 For n = 14, the a(14) = 8 valid compositions are:
%e A338271 14 = 2+2+2+2+2+3+1 and 2 = s(2+s(2+s(2+s(2+s(2+s(3+s(1)))))))
%e A338271 14 = 1+7+2+3+1     and 2 = s(1+s(7+s(2+s(3+s(1)))))
%e A338271 14 = 2+1+7+3+1     and 2 = s(2+s(1+s(7+s(3+s(1)))))
%e A338271 14 = 2+2+1+8+1     and 2 = s(2+s(2+s(1+s(8+s(1)))))
%e A338271 14 = 2+2+2+2+2+4   and 2 = s(2+s(2+s(2+s(2+s(2+s(4))))))
%e A338271 14 = 1+7+2+4       and 2 = s(1+s(7+s(2+s(4))))
%e A338271 14 = 2+1+7+4       and 2 = s(2+s(1+s(7+s(4))))
%e A338271 14 = 2+2+1+9       and 2 = s(2+s(2+s(1+s(9))))
%Y A338271 Cf. A000196, A338268.
%Y A338271 Cf. A098504, A101391, A143862, A238351, A238686, A325676. A335942.
%K A338271 nonn
%O A338271 1,4
%A A338271 _Peter Kagey_, Oct 19 2020