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%I #22 Dec 11 2020 22:56:26
%S 1,1,1,1,1,1,1,1,3,4,5,10,16,32,56,90,134,186,304,476,746,1308,2522,
%T 4845,9129,17260,32684,59908,106181,191779,337793,596689,1061991,
%U 1907311,3518903,6426672,12093858,22777645,42886411,81002076,151575988,285280108,529313088
%N Number of subsets of {1..n} whose cardinality is equal to the root-mean-square of the elements.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Root-Mean-Square.html">Root-Mean-Square</a>
%e a(12) = 10 subsets: {1}, {1, 2, 4, 5, 7, 11}, {1, 3, 5, 6, 8, 9}, {3, 4, 5, 6, 7, 9}, {1, 2, 3, 6, 7, 10, 12}, {2, 3, 4, 5, 8, 9, 12}, {2, 3, 6, 7, 8, 9, 10}, {3, 4, 5, 6, 7, 8, 12}, {1, 2, 5, 6, 9, 10, 11, 12} and {1, 4, 6, 7, 8, 9, 11, 12}.
%o (Python)
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def b(n, sos, c):
%o if n == 0:
%o if c>0:
%o if sos==c*c*c: return 1
%o return 0
%o return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
%o a = lambda n: b(n, 0, 0)
%o print([a(n) for n in range(1, 44)]) # _Michael S. Branicky_, Dec 10 2020
%Y Cf. A092670, A339454, A339484.
%K nonn
%O 1,9
%A _Ilya Gutkovskiy_, Dec 09 2020
%E a(24)-a(43) from _Michael S. Branicky_, Dec 09 2020
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