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A339569 Number of subsets of {1..n} whose cardinality is equal to the root-mean-square of the elements.

%I A339569 #23 Feb 16 2025 08:34:01
%S A339569 1,1,1,1,1,1,1,1,3,4,5,10,16,32,56,90,134,186,304,476,746,1308,2522,
%T A339569 4845,9129,17260,32684,59908,106181,191779,337793,596689,1061991,
%U A339569 1907311,3518903,6426672,12093858,22777645,42886411,81002076,151575988,285280108,529313088
%N A339569 Number of subsets of {1..n} whose cardinality is equal to the root-mean-square of the elements.
%H A339569 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Root-Mean-Square.html">Root-Mean-Square</a>
%e A339569 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 A339569 (Python)
%o A339569 from functools import lru_cache
%o A339569 @lru_cache(maxsize=None)
%o A339569 def b(n, sos, c):
%o A339569   if n == 0:
%o A339569     if c>0:
%o A339569       if sos==c*c*c: return 1
%o A339569     return 0
%o A339569   return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
%o A339569 a = lambda n: b(n, 0, 0)
%o A339569 print([a(n) for n in range(1, 44)]) # _Michael S. Branicky_, Dec 10 2020
%Y A339569 Cf. A092670, A339454, A339484.
%K A339569 nonn
%O A339569 1,9
%A A339569 _Ilya Gutkovskiy_, Dec 09 2020
%E A339569 a(24)-a(43) from _Michael S. Branicky_, Dec 09 2020