login
a(n) = Sum_{k=0..n} n CNIMPL k where CNIMPL = NOT(n) AND k is the bitwise logical converse non-implication operator (A102037).
3

%I #10 Dec 15 2021 08:00:24

%S 0,0,1,0,6,4,3,0,28,24,21,16,18,12,7,0,120,112,105,96,94,84,75,64,84,

%T 72,61,48,42,28,15,0,496,480,465,448,438,420,403,384,396,376,357,336,

%U 322,300,279,256,360,336,313,288,270,244,219,192,196,168,141,112

%N a(n) = Sum_{k=0..n} n CNIMPL k where CNIMPL = NOT(n) AND k is the bitwise logical converse non-implication operator (A102037).

%C The effect of NOT(n) AND k is to retain from k only those bits where n has a 0-bit. Conversely n AND k retains from k those bits where n has a 1-bit. Together they are all bits of k so that a(n) + A222423(n) = Sum_{k=0..n} k = n*(n+1)/2.

%F a(n) = (A001196(n) - n*(n+2))/4.

%F a(2*n) = 4*a(n) + n.

%F a(2*n+1) = 4*a(n).

%p with(Bits): cnimp := (n, k) -> And(Not(n), k):

%p seq(add(cnimp(n, k), k = 0..n), n = 0..59); # _Peter Luschny_, Dec 14 2021

%o (PARI) a(n) = (3*fromdigits(binary(n),4) - n^2 - 2*n)/4;

%Y Row sums of A102037.

%Y Cf. A001196 (bit doubling).

%Y Other sums: A222423 (AND), A350093 (OR), A224915 (XOR), A265736 (IMPL).

%K base,easy,nonn

%O 0,5

%A _Kevin Ryde_, Dec 14 2021