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

0, 2, 4, 12, 12, 22, 32, 56, 48, 58, 68, 100, 108, 142, 176, 240, 208, 210, 212, 252, 252, 294, 336, 424, 416, 458, 500, 596, 636, 734, 832, 992, 896, 866, 836, 876, 844, 886, 928, 1048, 1008, 1050, 1092, 1220, 1260, 1390, 1520, 1744, 1680, 1714, 1748, 1884, 1916

Offset: 0

Author

Peter Luschny, Sep 27 2024

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003986, A003987, A006582, A020522, A051933, A099027, A224915, A376585.

Programs

Maple

XOR := (n, k) -> Bits:-Xor(n, k):
a := n -> local k; add(XOR(k, n-k), k=0..n):
seq(a(n), n = 0..52);

Mathematica

(* Using definition *)
Table[Sum[BitXor[n - k, k], {k, 0, n}], {n, 0, 100}]
(* Using recurrence -- faster *)
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])];
Table[a[n], {n, 0, 100}] (* Paolo Xausa, Oct 01 2024 *)

PARI

a(n) = sum(k=0, n, bitxor(k, n-k)); \\ Michel Marcus, Sep 28 2024

Formula

a(n) = 2*A099027(n).

a(n) = 2*n + A006582(n).

a(2^n - 1) = 4^n - 2^n = A020522(n).

a(2^n) = 4^n - 2^n*(n - 1) = 2*A376585(n).

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.