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A306903 Sum over all partitions of n of the bitwise XOR of the parts.

%I A306903 #16 Mar 19 2019 04:34:45
%S A306903 0,1,2,7,8,19,26,61,70,126,146,270,308,519,604,1054,1222,1929,2208,
%T A306903 3454,3930,5862,6576,9833,11102,16052,17904,25752,28764,40479,44830,
%U A306903 62988,70188,97151,107662,148141,164710,223783,247380,334035,370406,495313,547000
%N A306903 Sum over all partitions of n of the bitwise XOR of the parts.
%H A306903 Alois P. Heinz, <a href="/A306903/b306903.txt">Table of n, a(n) for n = 0..1000</a>
%H A306903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise operation">Bitwise operation</a>
%H A306903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Commutative_property">Commutative property</a>
%H A306903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Identity_element">Identity element</a>
%H A306903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A306903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Truth_table">Truth table</a>
%F A306903 a(n) is odd <=> n in { A067567 }.
%p A306903 b:= proc(n, i, r) option remember; `if`(i<1, 0, (t->
%p A306903       `if`(i<n, b(n-i, min(i, n-i), t), 0)+
%p A306903       `if`(i=n, t, 0)+b(n, i-1, r))(Bits[Xor](i, r)))
%p A306903     end:
%p A306903 a:= n-> b(n$2, 0):
%p A306903 seq(a(n), n=0..45);
%Y A306903 Cf. A006906, A066186, A067567, A306884, A306895, A306901, A306902, A306925.
%K A306903 nonn,base
%O A306903 0,3
%A A306903 _Alois P. Heinz_, Mar 15 2019