A342448 Partial sums of A066194.

1, 3, 7, 10, 18, 25, 30, 36, 52, 67, 80, 94, 103, 113, 125, 136, 168, 199, 228, 258, 283, 309, 337, 364, 381, 399, 419, 438, 462, 485, 506, 528, 592, 655, 716, 778, 835, 893, 953, 1012, 1061, 1111, 1163, 1214, 1270, 1325, 1378, 1432, 1465, 1499, 1535, 1570

Offset

1,2

Comments

n^2/2 + n/2 <= a(n) <= (31/50)*n^2 + n/2. The lower and upper bounds are attained at n=2^k and n=5*2^k for k >= 0.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..16384

Formula

a(n) = A268836(n)/2 + n. - Kevin Ryde, Mar 12 2021

a(1) = 1; a(n) = [n == 0 (mod 2)]*(4*a(n/2) - n/2) + [n == 1 (mod 2)]*(2*a((n - 1)/2)+2*a((n + 1)/2)-(n-1)/2 - A010060(n)) where [] is an Iverson bracket

Maple

b:= proc(n) option remember; `if`(n<2, n,
      Bits[Xor](n, b(iquo(n, 2))))
    end:
a:= proc(n) a(n):= 1+`if`(n<2, 0, a(n-1)+b(n-1)) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 14 2021

Mathematica

a[1]=1;
a[n_/; EvenQ[n]]:= a[n] = 4a[n/2] - n/2;
a[n_/; OddQ[n]]:= a[n] = 2a[(n - 1)/2]+2a[(n + 1)/2]-(n-1)/2 - ThueMorse[n];
(* Second program: *)
b[n_] := If[n==0, 0, BitXor@@Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]];
A066194 = Table[b[n]+1, {n, 0, 60}];
A066194 // Accumulate (* Jean-François Alcover, Sep 10 2022 *)

Crossrefs

Cf. A066194, A268836, A007814, A000120, A010060, A000035.

Sequence in context: A171071 A265724 A352781 * A365931 A024330 A069153

Adjacent sequences: A342445 A342446 A342447 * A342449 A342450 A342451

Keyword

nonn,look

Author

John Erickson, Mar 12 2021

Status

approved