A255737 -id:A255737 - cp's OEIS frontend

A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

1, 5, 9, 21, 25, 37, 53, 85, 89, 101, 117, 149, 165, 201, 261, 341, 345, 357, 373, 405, 421, 457, 517, 597, 613, 649, 709, 793, 853, 965, 1173, 1365, 1369, 1381, 1397, 1429, 1445, 1481, 1541, 1621, 1637, 1673, 1733, 1817, 1877, 1989, 2197, 2389, 2405, 2441, 2501

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

Partial sums of A162793.
Also, total number of ON cells at stage n of the two-dimensional cellular automaton defined as follows: replace every "vertical" toothpick of length 2 with a centered unit square "ON" cell, so we have a cellular automaton which is similar to both A147562 and A169707 (this is the "one-step bishop" version). For the "one-step rook" version we use toothpicks of length sqrt(2), then rotate the structure 45 degrees and then replace every toothpick with a unit square "ON" cell. For the illustration of the sequence as a cellular automaton we now have three versions: the original version with toothpicks, the one-step rook version and one-step bishop version. Note that the last two versions refer to the standard ON cells in the same way as the two versions of A147562 and the two versions of A169707. It appears that the graph of this sequence lies between the graphs of A147562 and A169707. Also, it appears that this sequence shares infinitely many terms with both A147562 and A169707, see Formula section and Example section. - Omar E. Pol, Feb 20 2015
It appears that this is also a bisection (the odd terms) of A255747.

			From _Omar E. Pol_, Feb 18 2015: (Start)
Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
    1;
    5;
    9, 21;
   25, 37, 53, 85;
   89,101,117,149,165,201,261,341;
  345,357,373,405,421,457,517,597,613,649,709,793,853,965,1173,1365;
  ...
The right border gives the positive terms of A002450.
(End)
It appears that T(j,k) = A147562(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements of the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015

Michel Marcus, Table of n, a(n) for n = 1..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A002450, A048645, A139250, A139251, A147562, A153000, A159791, A159792, A160164, A160552, A162793, A162794, A162796, A162797, A169707, A255263, A255264, A255747.

It appears that a(n) = A147562(n) = A169707(n), if n is a term of A048645, otherwise A147562(n) < a(n) < A169707(n). - Omar E. Pol, Feb 20 2015
It appears that a(n) = (A169707(2n) - 1)/4 = A255747(2n-1). - Omar E. Pol, Mar 07 2015
a(n) = 1 + 4*A255737(n-1). - Omar E. Pol, Mar 08 2015

More terms from N. J. A. Sloane, Dec 28 2009

A256264 Partial sums of A256263.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 37, 42, 53, 70, 85, 86, 89, 94, 101, 106, 117, 134, 149, 154, 165, 182, 205, 234, 269, 310, 341, 342, 345, 350, 357, 362, 373, 390, 405, 410, 421, 438, 461, 490, 525, 566, 597, 602, 613, 630, 653, 682, 717, 758, 805, 858, 917, 982, 1053, 1130, 1213, 1302, 1365

Offset: 0

Omar E. Pol, Mar 30 2015

First differs from A255747 at a(27).

			Also, written as an irregular triangle 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,  42,  53,  70,  85;
86, 89, 94, 101, 106, 117, 134, 149, 154, 165, 182, 205, 234, 269,310,341;
...
It appears that the first column gives 0 together with the terms of A047849, hence the right border gives A002450.
It appears that this triangle at least shares with the triangles from the following sequences; A151920, A255737, A255747, A256249, the positive elements of the columns k, if k is a power of 2.
From _Omar E. Pol, Jan 02 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   42                  | | |_ _ _| | | |
13   53                  | |_ _ _ _ _| | |
14   70                  |_ _ _ _ _ _ _| |
15   85                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16   86                                  |_| | | |_ _  | |_ _ _ _ _ _  | |
17   89                                  |_ _| | |_  | | |_ _ _ _ _  | | |
18   94                                  |_ _ _| | | | | |_ _ _ _  | | | |
19  101                                  |_ _ _ _| | | | |_ _ _  | | | | |
20  106                                  | | |_ _ _| | | |_ _  | | | | | |
21  117                                  | |_ _ _ _ _| | |_  | | | | | | |
22  134                                  |_ _ _ _ _ _ _| | | | | | | | | |
23  149                                  |_ _ _ _ _ _ _ _| | | | | | | | |
24  154                                  | | | | | | |_ _ _| | | | | | | |
25  165                                  | | | | | |_ _ _ _ _| | | | | | |
26  182                                  | | | | |_ _ _ _ _ _ _| | | | | |
27  205                                  | | | |_ _ _ _ _ _ _ _ _| | | | |
28  234                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |
29  269                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
30  310                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
31  341                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the total number of cells in the first n regions of the diagram. A256263(n) gives the number of cells in the n-th region of the diagram.
(End)

Ivan Neretin, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A002450, A011782, A047849, A139250, A151920, A255737, A255747, A256249, A256258, A256260, A256261, A256263, A256265.

Accumulate@Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 5]] (* Ivan Neretin, Feb 14 2017 *)

a(n) = (A256260(n+1) - 1)/4.

A256249 Partial sums of A006257 (Josephus problem).

Original entry on oeis.org

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

Omar E. Pol, Mar 20 2015

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.

			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)

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

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

(* 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 *)

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

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

cp's OEIS Frontend

A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

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A256264 Partial sums of A256263.

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A256249 Partial sums of A006257 (Josephus problem).

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