A199111 -id:A199111 - cp's OEIS frontend

A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

0, 1, 3, 5, 9, 9, 9, 13, 25, 21, 9, 13, 25, 25, 25, 37, 73, 57, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 165, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 169, 25, 37, 73, 73, 73, 109, 217, 181, 73, 109, 217, 217, 217, 325, 649, 489, 9, 13, 25

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

Essentially the first differences of A160172.
For further information see the Applegate-Pol-Sloane paper, chapter 11: T-shaped toothpicks. See also the figure 16 in the mentioned paper. - Omar E. Pol, Nov 18 2011
The numbers n in increasing order such that the triple [n, n, n] can be found here, give A199111. [Observed by Omar E. Pol, Nov 18 2011. Confirmed by Alois P. Heinz, Nov 21 2011]

			From _Omar E. Pol_, Feb 09 2010: (Start)
If written as a triangle:
0;
1;
3;
5;
9,9;
9,13,25,21;
9,13,25,25,25,37,73,57;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,165;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109,217,181,73,109,217,217,217,325,649,489;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109...
(End)

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

David Applegate, The movie version
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.], which is also available at arXiv:1004.3036v2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160172, A160121, A160171, A161207, A161329, A172311.

wt[n_] := DigitCount[n, 2, 1];
a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := 2/3 (3^wt[n-1] + 3^wt[n-2]) + 1;
Table[a[n], {n, 0, 68}] (* Jean-François Alcover, Aug 18 2018, after N. J. A. Sloane *)

a(n) = (2/3)*(3^wt(n-1) + 3^wt(n-2))+1 (where wt is A000120), for n >= 3. - N. J. A. Sloane, Jan 01 2010

More terms from N. J. A. Sloane, Jan 01 2010

A094617 Triangular array T of numbers generated by these rules: 2 is in T; and if x is in T, then 2x-1 and 3x-2 are in T.

Original entry on oeis.org

2, 3, 4, 5, 7, 10, 9, 13, 19, 28, 17, 25, 37, 55, 82, 33, 49, 73, 109, 163, 244, 65, 97, 145, 217, 325, 487, 730, 129, 193, 289, 433, 649, 973, 1459, 2188, 257, 385, 577, 865, 1297, 1945, 2917, 4375, 6562, 513, 769, 1153, 1729, 2593, 3889, 5833, 8749, 13123, 19684

Offset: 1

Clark Kimberling, May 14 2004

To obtain row n from row n-1, apply 2x-1 to each x in row n-1 and then put 1+3^n at the end. Or, instead, apply 3x-2 to each x in row n-1 and then put 1+2^n at the beginning.
From Lamine Ngom, Feb 10 2021: (Start)
Triangle read by diagonals provides all the sequences of the form 1+2^(k-1)*3^n, where k is the k-th diagonal.
For instance, the terms of the first diagonal form the sequence 2, 4, 10, 28, ..., i.e., 1+3^n (A034472).
The 2nd diagonal leads to the sequence 3, 7, 19, 55, ..., i.e., 1+2*3^n (A052919).
The 3rd diagonal is the sequence 5, 13, 37, 109, ..., i.e., 1+4*3^n (A199108).
And for k = 4, we obtain the sequence 9, 25, 73, 217, ..., i.e., 1+8*3^n (A199111). (End)

			Rows of this triangle begin:
    2;
    3,   4;
    5,   7,   10;
    9,  13,   19,   28;
   17,  25,   37,   55,   82;
   33,  49,   73,  109,  163,  244;
   65,  97,  145,  217,  325,  487,  730;
  129, 193,  289,  433,  649,  973, 1459, 2188;
  257, 385,  577,  865, 1297, 1945, 2917, 4375,  6562;
  513, 769, 1153, 1729, 2593, 3889, 5833, 8749, 13123, 19684;
  ...

Ivan Neretin, Table of n, a(n) for n = 1..5151

Cf. A094616, A094618, A138247.

Cf. A034472, A052919, A199108, A199111.

FoldList[Append[2 #1 - 1, 1 + 3^#2] &, {2}, Range[9]] // Flatten (* Ivan Neretin, Mar 30 2016 *)

When offset is zero, then the first term is T(0,0) = 2, and
T(n,0) = 1 + 2^n = A000051(n),
T(n,n) = 1 + 3^n = A048473(n),
T(2n,n) = 1 + 6^n = A062394(n).
Row sums = A094618.
a(n) = A036561(n-1) + 1. - Filip Zaludek, Nov 19 2016

A198644 a(n) = 8*3^n - 1.

Original entry on oeis.org

7, 23, 71, 215, 647, 1943, 5831, 17495, 52487, 157463, 472391, 1417175, 4251527, 12754583, 38263751, 114791255, 344373767, 1033121303, 3099363911, 9298091735, 27894275207, 83682825623, 251048476871, 753145430615, 2259436291847, 6778308875543, 20334926626631, 61004779879895

Offset: 0

Vincenzo Librandi, Oct 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A199111.

Magma
```
[8*3^n-1: n in [0..30]];
```

8*3^Range[0,30]-1 (* or *) LinearRecurrence[{4,-3},{7,23},30] (* Harvey P. Dale, Jun 16 2024 *)

a(n) = 3*a(n-1) + 2, a(0)=7.
G.f.: (7-5*x)/((3*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011
From Elmo R. Oliveira, May 07 2025: (Start)
E.g.f.: exp(x)*(8*exp(2*x) - 1).
a(n) = 4*a(n-1) - 3*a(n-2).
a(n) = A199111(n) - 2. (End)

cp's OEIS Frontend

A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

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A094617 Triangular array T of numbers generated by these rules: 2 is in T; and if x is in T, then 2x-1 and 3x-2 are in T.

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A198644 a(n) = 8*3^n - 1.

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Magma

Mathematica

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