A256249 Partial sums of A006257 (Josephus problem).

0, 1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 37, 46, 57, 70, 85, 86, 89, 94, 101, 110, 121, 134, 149, 166, 185, 206, 229, 254, 281, 310, 341, 342, 345, 350, 357, 366, 377, 390, 405, 422, 441, 462, 485, 510, 537, 566, 597, 630, 665, 702, 741, 782, 825, 870, 917, 966, 1017, 1070, 1125, 1182, 1241, 1302, 1365, 1366, 1369, 1374

Offset

0,3

Comments

Also total number of ON states after n generations in one of the four wedges of the one-step rook version (or in one of the four quadrants of the one-step bishop version) of the cellular automaton of A256250.

A006257 gives the number of cells turned ON at n-th stage.

First differs from A255747 at a(11).

First differs from A169779 at a(10).

It appears that the odd terms (a bisection) give A256250.

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

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. 37, 41.

Yuri Nikolayevsky and Ioannis Tsartsaflis, Cohomology of N-graded Lie algebras of maximal class over Z_2, arXiv:1512.87676 [math.RA], (2016), page 6.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

Index entries for sequences related to the Josephus Problem

Formula

a(n) = (A256250(n+1) - 1)/4.

Example

Written as an irregular triangle T(n,k), k >= 1, in which the row lengths are the terms of A011782 the sequence begins:
   0;
   1;
   2,  5;
   6,  9, 14, 21;
  22, 25, 30, 37, 46, 57, 70, 85;
  86, 89, 94,101,110,121,134,149,166,185,206,229,254,281,310,341;
  ...
Right border, a(2^m-1), gives A002450(m) for m >= 0.
a(2^m-2) = A203241(m) for m >= 2.
It appears that this triangle at least shares with the triangles from the following sequences; A151920, A255737, A255747, the positive elements of the columns k, if k is a power of 2.
From Omar E. Pol, Jan 03 2016: (Start)
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n    a(n)                 Compact diagram
---------------------------------------------------------------------------
0     0     _
1     1    |_|_ _
2     2      |_| |
3     5      |_ _|_ _ _ _
4     6          |_| | | |
5     9          |_ _| | |
6    14          |_ _ _| |
7    21          |_ _ _ _|_ _ _ _ _ _ _ _
8    22                  |_| | | | | | | |
9    25                  |_ _| | | | | | |
10   30                  |_ _ _| | | | | |
11   37                  |_ _ _ _| | | | |
12   46                  |_ _ _ _ _| | | |
13   57                  |_ _ _ _ _ _| | |
14   70                  |_ _ _ _ _ _ _| |
15   85                  |_ _ _ _ _ _ _ _|
.
a(n) is also the total number of cells in the first n regions of the diagram. A006257(n) gives the number of cells in the n-th region of the diagram.
(End)

Mathematica

(* Based on Birkas Gyorgy's code in A006257 *)
Accumulate[Prepend[Flatten[Table[Range[1, 2^n-1, 2], {n, 0, 7}]], 0]] (* Ivan N. Ianakiev, Mar 30 2015 *)

Prog

(PARI) a(n)=n++; b=#binary(n>>1); (4^b-1)/3+(n-2^b)^2 \\ David A. Corneth, Mar 21 2015

Crossrefs

Cf. A002450, A006257, A011782, A139250, A151920, A169779, A255737, A255747, A256250, A256251, A266540.

Sequence in context: A226793 A255747 A256264 * A169779 A326118 A301791

Adjacent sequences: A256246 A256247 A256248 * A256250 A256251 A256252

Keyword

nonn,look

Author

Omar E. Pol, Mar 20 2015

Status

approved