A006583 a(n) = Sum_{k=1..n-1} (k OR n-k). (Formerly M4069)

1, 6, 8, 16, 25, 42, 44, 56, 69, 94, 108, 136, 165, 210, 208, 224, 241, 278, 296, 336, 377, 442, 460, 504, 549, 622, 668, 744, 821, 930, 912, 928, 945, 998, 1016, 1072, 1129, 1226, 1244, 1304

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

Michael De Vlieger, Table of n, a(n) for n = 2..8192

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, p. 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^2(4t^2+6t+1)/(1+t)^2, t=x^2^k). - Ralf Stephan, Feb 12 2003

a(0)=a(1)=0, a(2n) = 2a(n)+2a(n-1)+5n-4, a(2n+1) = 4a(n)+6n. - Ralf Stephan, Oct 09 2003

a(n) = 2*(Sum_{k=1..floor((n-1)/2)} k OR n-k) + m where m is 0 if n is odd and n/2 otherwise. - Chai Wah Wu, May 07 2023

Mathematica

Table[Sum[BitOr[k, n-k], {k, n-1}], {n, 2, 50}] (* Harvey P. Dale, Dec 05 2020 *)

Prog

(PARI) a(n)=sum(k=1, n-1, bitor(k, n-k)) \\ Charles R Greathouse IV, Aug 11 2017
(Python)
def A006583(n): return (sum(k|n-k for k in range(1, n+1>>1))<<1)+(0 if n&1 else n>>1) # Chai Wah Wu, May 07 2023

Crossrefs

Antidiagonal sums of array A003986.

Sequence in context: A315940 A048228 A067543 * A011989 A270821 A139452

Adjacent sequences: A006580 A006581 A006582 * A006584 A006585 A006586

Keyword

nonn,base

Author

N. J. A. Sloane

Status

approved