A006582 a(n) = Sum_{k=1..n-1} k XOR n-k. (Formerly M4053)

0, 6, 4, 12, 20, 42, 32, 40, 48, 78, 84, 116, 148, 210, 176, 176, 176, 214, 212, 252, 292, 378, 368, 408, 448, 542, 580, 676, 772, 930, 832, 800, 768, 806, 772, 812, 852, 970, 928, 968, 1008, 1134, 1172, 1300, 1428, 1650, 1584, 1616, 1648, 1782, 1812, 1948

Offset

2,2

References

Marc LeBrun, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

R. J. Mathar, Table of n, a(n) for n = 2..1000

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 38-39.

M. Le Brun, Email to N. J. A. Sloane, Jul 1991

Formula

G.f.: 1/(1-x)^2 * Sum_{k>=0} 2^k * t^3(4t+6)/(1+t)^2, t=x^2^k. - Ralf Stephan, Feb 12 2003

a(0) = a(1) = 0, a(2n) = 2a(n) + 2a(n-1) + 4n - 4, a(2n+1) = 4a(n) + 6n. - Ralf Stephan, Oct 09 2003

a(n) = 2*(Sum_{k=1..floor((n-1)/2)} k XOR n-k). - Chai Wah Wu, May 07 2023

Maple

A006582 := proc(n)
    add(A003987(k, n-k), k=1..n-1) ;
end proc: # R. J. Mathar, Apr 17 2013

Mathematica

Array[Sum[BitXor[k, # - k], {k, # - 1}] &, 52, 2] (* Michael De Vlieger, Oct 27 2022 *)

Prog

(PARI) a(n)=if(n<2, 0, if(n%2==0, 2*a(n/2)+2*a(n/2-1)+4*(n/2-1), 4*a((n-1)/2)+6*((n-1)/2)))
(PARI) a(n)=sum(k=1, n-1, bitxor(k, n-k)) \\ Charles R Greathouse IV, Aug 11 2017
(Python)
def A006582(n): return sum(k^n-k for k in range(1, n+1>>1))<<1 # Chai Wah Wu, May 07 2023

Crossrefs

Antidiagonal sums of array A003987.

Sequence in context: A141270 A040032 A239394 * A263586 A180497 A213038

Adjacent sequences: A006579 A006580 A006581 * A006583 A006584 A006585

Keyword

nonn

Author

N. J. A. Sloane

Status

approved