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A126024 Number of subsets of {1,2,3,...,n} whose sum is a square integer (including the empty subset).

%I A126024 #26 Nov 03 2023 11:06:56
%S A126024 1,2,2,3,5,7,12,20,34,60,106,190,346,639,1183,2204,4129,7758,14642,
%T A126024 27728,52648,100236,191294,365827,700975,1345561,2587057,4981567,
%U A126024 9605777,18546389,35851756,69382558,134414736,260658770,505941852,982896850
%N A126024 Number of subsets of {1,2,3,...,n} whose sum is a square integer (including the empty subset).
%H A126024 Alois P. Heinz, <a href="/A126024/b126024.txt">Table of n, a(n) for n = 0..990</a> (terms n=1..100 from T. D. Noe)
%e A126024 The subsets of {1,2,3,4,5} that sum to a square are {}, {1}, {1,3}, {4}, {2,3,4}, {1,3,5} and {4,5}. Thus a(5)=7.
%p A126024 b:= proc(n, i) option remember; (m->
%p A126024       `if`(n=0 or n=m, 1, `if`(n<0 or n>m, 0, b(n, i-1)+
%p A126024       `if`(i>n, 0, b(n-i, i-1)))))(i*(i+1)/2)
%p A126024     end:
%p A126024 a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+
%p A126024       add(b(j^2-n, n-1), j=isqrt(n)..isqrt(n*(n+1)/2)))
%p A126024     end:
%p A126024 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 02 2017
%t A126024 g[n_] := Block[{p = Product[1 + z^i, {i, n}]},Sum[Boole[IntegerQ[Sqrt[k]]]*Coefficient[p, z, k], {k, 0, n*(n + 1)/2}]];Array[g, 35] (* _Ray Chandler_, Mar 05 2007 *)
%o A126024 (Haskell)
%o A126024 import Data.List (subsequences)
%o A126024 a126024 = length . filter ((== 1) . a010052 . sum) .
%o A126024                           subsequences . enumFromTo 1
%o A126024 -- _Reinhard Zumkeller_, Feb 22 2012, Oct 27 2010
%Y A126024 Cf. A053632, A127542.
%Y A126024 Cf. A181522. - _Reinhard Zumkeller_, Oct 27 2010
%Y A126024 Cf. A010052, A284250.
%Y A126024 Row sums of A281871.
%K A126024 nonn
%O A126024 0,2
%A A126024 _John W. Layman_, Feb 27 2007
%E A126024 Extended by _Ray Chandler_, Mar 05 2007
%E A126024 a(0)=1 prepended by _Alois P. Heinz_, Jan 30 2017