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A375551 a(n) = Sum_{k=0..n} k XOR n-k, where XOR is the bitwise exclusive disjunction. Row sums of A003987.

%I A375551 #26 Oct 01 2024 12:05:05
%S A375551 0,2,4,12,12,22,32,56,48,58,68,100,108,142,176,240,208,210,212,252,
%T A375551 252,294,336,424,416,458,500,596,636,734,832,992,896,866,836,876,844,
%U A375551 886,928,1048,1008,1050,1092,1220,1260,1390,1520,1744,1680,1714,1748,1884,1916
%N A375551 a(n) = Sum_{k=0..n} k XOR n-k, where XOR is the bitwise exclusive disjunction. Row sums of A003987.
%H A375551 Paolo Xausa, <a href="/A375551/b375551.txt">Table of n, a(n) for n = 0..10000</a>
%F A375551 a(n) = 2*A099027(n).
%F A375551 a(n) = 2*n + A006582(n).
%F A375551 a(2^n - 1) = 4^n - 2^n = A020522(n).
%F A375551 a(2^n) = 4^n - 2^n*(n - 1) = 2*A376585(n).
%F A375551 Recurrence: a(0) = 0; a(2*n) = 2*(a(n) + a(n-1)); a(2*n+1) = 2*(2*a(n) + n + 1). - _Paolo Xausa_, Oct 01 2024, derived from recurrence in A099027.
%p A375551 XOR := (n, k) -> Bits:-Xor(n, k):
%p A375551 a := n -> local k; add(XOR(k, n-k), k=0..n):
%p A375551 seq(a(n), n = 0..52);
%t A375551 (* Using definition *)
%t A375551 Table[Sum[BitXor[n - k, k], {k, 0, n}], {n, 0, 100}]
%t A375551 (* Using recurrence -- faster *)
%t A375551 a[0] = 0; a[n_] := a[n] = If[OddQ[n], 4*a[(n-1)/2] + n + 1, 2*(a[n/2] + a[n/2-1])];
%t A375551 Table[a[n], {n, 0, 100}] (* _Paolo Xausa_, Oct 01 2024 *)
%o A375551 (PARI) a(n) = sum(k=0, n, bitxor(k, n-k)); \\ _Michel Marcus_, Sep 28 2024
%Y A375551 Cf. A003986, A003987, A006582, A020522, A051933, A099027, A224915, A376585.
%K A375551 nonn
%O A375551 0,2
%A A375551 _Peter Luschny_, Sep 27 2024