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A338268 Irregular table read by rows: T(n,k) is the number of compositions of n, b_1 + ... + b_t = n, such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) = k; 1 <= k <= A000196(n).

%I A338268 #32 Nov 06 2020 07:51:22
%S A338268 1,0,0,0,2,0,0,0,2,0,0,0,2,0,0,2,0,4,0,0,0,2,0,6,0,0,0,2,0,8,0,0,0,4,
%T A338268 0,12,0,2,0,0,6,0,0,18,0,2,0,0,8,0,0,26,0,2,0,0,14,0,0,40,0,4,0,0,20,
%U A338268 0,0,60,0,6,0,0,28,0,2,0,88,0,8,0
%N A338268 Irregular table read by rows:  T(n,k) is the number of compositions of n, b_1 + ... + b_t = n, such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) = k; 1 <= k <= A000196(n).
%C A338268 For any fixed c, T(x^2 + c, x) = T(y^2 + c, y) for sufficiently large integers x and y. See A338286.
%C A338268 T(n,k) <= A338286(n - k^2).
%H A338268 Peter Kagey, <a href="/A338268/b338268.txt">Table of n, a(n) for n = 1..5150</a> (first 400 rows)
%H A338268 Code Golf Stack Exchange user Bubbler, <a href="https://codegolf.stackexchange.com/a/213851/53884">The square root of the square root of the square root of the...</a>
%H A338268 Peter Kagey, <a href="/A338268/a338268.txt">Table of first 15^2 = 225 rows</a>.
%F A338268 T(n,1) = 0 for n > 1.
%F A338268 T(n,k) = 0 if n + k is odd.
%e A338268 Table begins:
%e A338268   n\k| 1  2 3 4
%e A338268   ---+---------
%e A338268    1 | 1
%e A338268    2 | 0
%e A338268    3 | 0
%e A338268    4 | 0  2
%e A338268    5 | 0  0
%e A338268    6 | 0  2
%e A338268    7 | 0  0
%e A338268    8 | 0  2
%e A338268    9 | 0  0 2
%e A338268   10 | 0  4 0
%e A338268   11 | 0  0 2
%e A338268   12 | 0  6 0
%e A338268   13 | 0  0 2
%e A338268   14 | 0  8 0
%e A338268   15 | 0  0 4
%e A338268   16 | 0 12 0 2
%e A338268 The T(15,3) = 4 compositions of 15 whose iterated sum of square roots equals 3 are:
%e A338268 7 + 2 + 2 + 3 + 1,
%e A338268 7 + 2 + 2 + 4,
%e A338268 6 + 8 + 1, and
%e A338268 6 + 9.
%Y A338268 Cf. A000196, A338271, A338286.
%K A338268 nonn,tabf
%O A338268 1,5
%A A338268 _Peter Kagey_, Oct 19 2020