A266540 Partial sums of A266539.

0, 0, 1, 2, 3, 4, 7, 10, 11, 12, 15, 18, 23, 28, 35, 42, 43, 44, 47, 50, 55, 60, 67, 74, 83, 92, 103, 114, 127, 140, 155, 170, 171, 172, 175, 178, 183, 188, 195, 202, 211, 220, 231, 242, 255, 268, 283, 298, 315, 332, 351, 370, 391, 412, 435, 458, 483, 508, 535, 562, 591, 620, 651, 682, 683, 684, 687, 690, 695, 700

Offset: 1

Author

Omar E. Pol, Jan 02 2016

Keywords

• nonn

Comments

Also A266535 and twice the terms of A256249 interleaved, or in other words A266535 and A266538 interleaved.

It appears that this sequence has a fractal (or fractal-like) behavior.

First differs from both A266510 and A266530 at a(25), with which it shares infinitely many terms.

For an illustration of initial terms consider the diagram of A256249 in the fourth quadrant of the square grid together with a reflected copy in the second quadrant.

Also the third sequence of Betti numbers of the Lie algebra m_0(n) over Z_2. See the Nikolayevsky-Tsartsaflis paper, pages 2 and 6. Note that a(n) is denoted by b_3(m_0(n)).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Yuri Nikolayevsky and Ioannis Tsartsaflis, Cohomology of N-graded Lie algebras of maximal class over Z_2, arXiv:1512.87676 [math.RA], (2016); see pp. 2 and 6.
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A006257 (Josephus problem), A256249, A266535, A266510, A266530, A266538, A266539.

Programs

Maple

A006257[0]:=0: for n from 1 to 100 do A006257[n]:=(A006257[n-1]+1) mod n +1: end do:
ListTools:-PartialSums([seq(A006257[i]$2,i=0..100)]); # Robert Israel, Jan 13 2016

Mathematica

Join[{0, 0}, Table[{k, k}, {n, 1, 6}, {k, 1, 2^n-1, 2}] // Flatten] // Accumulate (* Jean-François Alcover, Sep 19 2018 *)

Formula

a(2n-1) = A266535(n).

a(2n) = 2 * A256249(n-1) = A266538(n-1).

a(n) = (a(n-1) + a(n+1))/2, if n is an odd number greater than 1.

G.f.: (x^3+x^5)/(1-2*x+2*x^3-x^4) - x*(1-x)^(-2)*Sum_{k>=1} 2^k*x^(2^(1+k)). - Robert Israel, Jan 13 2016