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
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
KEYWORD
nonn,look
AUTHOR
Omar E. Pol, Mar 20 2015
STATUS
approved