A160552 -id:A160552 - cp's OEIS frontend

A151574 a(0)=1, a(1)=2; a(2^i + j) = a(j) + 2*a(j+1) for 0 <= j < 2^i.

1, 2, 5, 12, 5, 12, 29, 22, 5, 12, 29, 22, 29, 70, 73, 32, 5, 12, 29, 22, 29, 70, 73, 32, 29, 70, 73, 80, 169, 216, 137, 42, 5, 12, 29, 22, 29, 70, 73, 32, 29, 70, 73, 80, 169, 216, 137, 42, 29, 70, 73, 80, 169, 216, 137, 90, 169, 216, 233, 418, 601, 490, 221, 52, 5, 12, 29, 22, 29

Offset: 0

N. J. A. Sloane, May 25 2009

nonn

Equals 2*A151572 + A151703.

Ivan Neretin, Table of n, a(n) for n = 0..8191
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

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.

a = {1, 2}; Do[AppendTo[a, a[[j]] + 2 a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* Ivan Neretin, Jun 28 2017 *)

A151704 a(0)=1, a(1)=0; a(2^i+j) = 2*a(j) + a(j+1) for 0 <= j < 2^i.

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 6, 6, 2, 2, 6, 6, 6, 10, 18, 14, 2, 2, 6, 6, 6, 10, 18, 14, 6, 10, 18, 18, 22, 38, 50, 30, 2, 2, 6, 6, 6, 10, 18, 14, 6, 10, 18, 18, 22, 38, 50, 30, 6, 10, 18, 18, 22, 38, 50, 34, 22, 38, 54, 58, 82, 126, 130, 62, 2, 2, 6, 6, 6, 10, 18, 14, 6, 10, 18, 18, 22, 38, 50, 30, 6

Offset: 0

N. J. A. Sloane, Jun 06 2009

nonn

Ivan Neretin, Table of n, a(n) for n = 0..8191
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

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.

Maple
```
See A151702 for Maple code.
```

a = {1, 0}; Do[AppendTo[a, 2 a[[j]] + a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* Ivan Neretin, Jul 04 2017 *)

A151705 a(0)=0, a(1)=1; a(2^i+j) = 2a(j) + 2a(j+1) for 0 <= j < 2^i.

Original entry on oeis.org

0, 1, 2, 6, 2, 6, 16, 16, 2, 6, 16, 16, 16, 44, 64, 36, 2, 6, 16, 16, 16, 44, 64, 36, 16, 44, 64, 64, 120, 216, 200, 76, 2, 6, 16, 16, 16, 44, 64, 36, 16, 44, 64, 64, 120, 216, 200, 76, 16, 44, 64, 64, 120, 216, 200, 104, 120, 216, 256, 368, 672, 832, 552, 156, 2, 6, 16, 16, 16

Offset: 0

N. J. A. Sloane, Jun 06 2009

nonn

Ivan Neretin, Table of n, a(n) for n = 0..8191
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

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.

Maple
```
See A151702 for Maple code.
```

a = {0, 1}; Do[AppendTo[a, 2 a[[j]] + 2 a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* Ivan Neretin, Jul 04 2017 *)

A151706 a(0)=1, a(1)=0; a(2^i+j) = 2a(j) + 2a(j+1) for 0 <= j < 2^i.

Original entry on oeis.org

1, 0, 2, 4, 2, 4, 12, 12, 2, 4, 12, 12, 12, 32, 48, 28, 2, 4, 12, 12, 12, 32, 48, 28, 12, 32, 48, 48, 88, 160, 152, 60, 2, 4, 12, 12, 12, 32, 48, 28, 12, 32, 48, 48, 88, 160, 152, 60, 12, 32, 48, 48, 88, 160, 152, 80, 88, 160, 192, 272, 496, 624, 424, 124, 2, 4, 12, 12, 12, 32, 48

Offset: 0

N. J. A. Sloane, Jun 06 2009

nonn

Ivan Neretin, Table of n, a(n) for n = 0..8191
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

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.

Maple
```
See A151702 for Maple code.
```

a = {1, 0}; Do[AppendTo[a, 2 a[[j]] + 2 a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* Ivan Neretin, Jul 04 2017 *)

A151707 a(0)=1, a(1)=1; a(2^i+j) = 2a(j) + 2a(j+1) for 0 <= j < 2^i.

Original entry on oeis.org

1, 1, 4, 10, 4, 10, 28, 28, 4, 10, 28, 28, 28, 76, 112, 64, 4, 10, 28, 28, 28, 76, 112, 64, 28, 76, 112, 112, 208, 376, 352, 136, 4, 10, 28, 28, 28, 76, 112, 64, 28, 76, 112, 112, 208, 376, 352, 136, 28, 76, 112, 112, 208, 376, 352, 184, 208, 376, 448, 640, 1168, 1456, 976, 280

Offset: 0

N. J. A. Sloane, Jun 06 2009

nonn

Equals A151705 + A151706.

Ivan Neretin, Table of n, a(n) for n = 0..8191
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

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.

Maple
```
See A151702 for Maple code.
```

a = {1, 1}; Do[AppendTo[a, 2 a[[j]] + 2 a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* Ivan Neretin, Jul 04 2017 *)

A151708 a(0)=1, a(1)=2; a(2^i+j)=2a(j)+2a(j+1) for 0 <= j < 2^i.

Original entry on oeis.org

1, 2, 6, 16, 6, 16, 44, 44, 6, 16, 44, 44, 44, 120, 176, 100, 6, 16, 44, 44, 44, 120, 176, 100, 44, 120, 176, 176, 328, 592, 552, 212, 6, 16, 44, 44, 44, 120, 176, 100, 44, 120, 176, 176, 328, 592, 552, 212, 44, 120, 176, 176, 328, 592, 552, 288, 328, 592, 704, 1008, 1840, 2288

Offset: 0

N. J. A. Sloane, Jun 06 2009

nonn

Equals 2*A151705+A151706.

Ivan Neretin, Table of n, a(n) for n = 0..8191
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

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.

Maple
```
See A151702 for Maple code.
```

a = {1, 2}; Do[AppendTo[a, 2 a[[j]] + 2 a[[j + 1]]], {i, 6}, {j, 2^i}]; a (* Ivan Neretin, Jul 04 2017 *)

A169708 First differences of A169707.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 20, 44, 68, 60, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 20, 44, 68, 76, 84, 156, 196, 140, 84, 156, 212, 236, 324, 508, 516, 252, 4, 12, 20, 28, 20

Offset: 0

N. J. A. Sloane, Apr 17 2010

			From _Omar E. Pol_, Feb 13 2015: (Start)
Written as an irregular triangle in which row lengths are 1,1,2,4,8,16,32,... the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124,20,44,68,76,84,156,196,140,84,156,212,236,324,508,516,252;
It appears that the row sums give A000302.
It appears that the right border gives A173033.
(End)

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000302, A011782, A139251, A151548, A152980, A153006, A160164, A160552, A169707 (partial sums), A170903, A173033, A253088.

It appears that a(n) = 4*A160552(n), n >= 1. - Omar E. Pol, Feb 13 2015

Initial 1 added by Omar E. Pol, Feb 13 2015

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89

Offset: 0

Omar E. Pol, Mar 30 2015

Partial sums give A256264.
First differs from A160552 at a(27).
Appears to be a canonical sequence partially related to the cellular automata of A139250, A147562, A162795, A169707, A255366, A256250. See also A256264 and A256260.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,3;
1,3,5,7;
1,3,5,7,5,11,17,15;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31,5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
...
Right border gives A000225.
Apart from the initial 0 the row sums give A000302.
Rows converge to A256258.
.
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n   a(n)                 Compact diagram
---------------------------------------------------------------------------
0    0     _
1    1    |_|_ _
2    1      |_| |
3    3      |_ _|_ _ _ _
4    1          |_| | | |
5    3          |_ _| | |
6    5          |_ _ _| |
7    7          |_ _ _ _|_ _ _ _ _ _ _ _
8    1                  |_| | | |_ _  | |
9    3                  |_ _| | |_  | | |
10   5                  |_ _ _| | | | | |
11   7                  |_ _ _ _| | | | |
12   5                  | | |_ _ _| | | |
13  11                  | |_ _ _ _ _| | |
14  17                  |_ _ _ _ _ _ _| |
15  15                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16   1                                  |_| | | |_ _  | |_ _ _ _ _ _  | |
17   3                                  |_ _| | |_  | | |_ _ _ _ _  | | |
18   5                                  |_ _ _| | | | | |_ _ _ _  | | | |
19   7                                  |_ _ _ _| | | | |_ _ _  | | | | |
20   5                                  | | |_ _ _| | | |_ _  | | | | | |
21  11                                  | |_ _ _ _ _| | |_  | | | | | | |
22  17                                  |_ _ _ _ _ _ _| | | | | | | | | |
23  15                                  |_ _ _ _ _ _ _ _| | | | | | | | |
24   5                                  | | | | | | |_ _ _| | | | | | | |
25  11                                  | | | | | |_ _ _ _ _| | | | | | |
26  17                                  | | | | |_ _ _ _ _ _ _| | | | | |
27  23                                  | | | |_ _ _ _ _ _ _ _ _| | | | |
28  29                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |
29  35                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
30  41                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
31  31                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
A256264(n) gives the total number of cells after n-th stage.

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. A000225, A000302, A011782, A038573, A006257, A016969, A139251, A160552, A256250, A256258, A256260, A256261, A256264, A256265.

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

Terms a(95) to a(98) fixed by Ivan Neretin, Feb 14 2017

A162956 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 7, 13, 1, 4, 7, 13, 7, 19, 34, 40, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 364, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46

Offset: 0

Gary W. Adamson, Jul 18 2009

nonn

2^n term triangle by rows, analogous to A160552 but multiplier is "3" instead of "2"
Row sums = powers of 5: (1, 5, 25, 125, 625,...).
Rows tend to A162957, obtained by taking (1, 3, 0, 0, 0,...) * A162956.

			The triangle begins:
0;
1;
1, 4;
1, 4, 7, 13;
1, 4, 7, 13, 7, 19, 34, 40;
1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121;
...
Row 4 = (1, 4, 7, 13, 7, 19, 34, 40): brings down (1, 4, 7, 13) then 7 = 3*1 + 4, 19 = 3*4 + 7, 34 = 3*7 + 13, 40 = 3*13 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..16383
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. A162957, A162958, A160552.

Cf. A170838-A170852, A170854-A170872.

a:= proc(n) option remember; `if`(n<2, n,
      (j-> 3*a(j)+a(j+1))(n-2^ilog2(n)))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 28 2017

row[0] = {0}; row[1] = {1}; row[n_] := row[n] = Join[row[n-1], 3 row[n-1] + Append[Rest[row[n-1]], 1]]; Table[row[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 13 2017 *)

Follows the same analogous procedure as A160552 but multiplier is 3 instead of 2. (n+1)-th row brings down n-th row and appends to the right and equal number of terms following the rules: from left to right,let a = last term, b = current term, c = next term. Then c = 3*a + b except for the rightmost term = 3*a + 1.

Edited with more terms by N. J. A. Sloane, Jan 02 2010

A162958 Equals A162956 convolved with (1, 3, 3, 3, ...).

Original entry on oeis.org

1, 4, 10, 19, 25, 40, 67, 94, 100, 115, 142, 175, 208, 280, 388, 469, 475, 490, 517, 550, 583, 655, 763, 850, 883, 955, 1069, 1201, 1372, 1696, 2101, 2344, 2350, 2365, 2392, 2425, 2458, 2530, 2638, 2725, 2758, 2830, 2944, 3076, 3247, 3571, 3976, 4225, 4258

Offset: 1

Gary W. Adamson, Jul 18 2009

nonn

Can be considered a toothpick sequence for N=3, following rules analogous to those in A160552 (= special case of "A"), A151548 = special case "B", and the toothpick sequence A139250 (N=2) = special case "C".
To obtain the infinite set of toothpick sequences, (N = 2, 3, 4, ...), replace the multiplier "2" in A160552 with any N, getting a triangle with 2^n terms. Convolve this A sequence with (1, N, 0, 0, 0, ...) = B such that row terms of A triangles converge to B.
Then generalized toothpick sequences (C) = A convolved with (1, N, N, N, ...).
Examples: A160552 * (1, 2, 0, 0, 0,...) = a B-type sequence A151548.
A160552 * (1, 2, 2, 2, 2,...) = the toothpick sequence A139250 for N=2.
A162956 is analogous to A160552 but replaces "2" with the multiplier "3".
Row terms of A162956 tend to A162957 = (1, 3, 0, 0, 0, ...) * A162956.
Toothpick sequence for N = 3 = A162958 = A162956 * (1, 3, 3, 3, ...).
Row sums of "A"-type triangles = powers of (N+2); since row sums of A160552 = (1, 4, 16, 64, ...), while row sums of A162956 = (1, 5, 25, 125, ...).
Is there an illustration of this sequence using toothpicks? - Omar E. Pol, Dec 13 2016

Alois P. Heinz, Table of n, a(n) for n = 1..16384
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. A139250, A152548, A160552, A162956, A163267.

Third diagonal of A163311.

b:= proc(n) option remember; `if`(n<2, n,
      (j-> 3*b(j)+b(j+1))(n-2^ilog2(n)))
    end:
a:= proc(n) option remember;
      `if`(n=0, 0, a(n-1)+2*b(n-1)+b(n))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 28 2017

b[n_] := b[n] = If[n<2, n, Function[j, 3*b[j]+b[j+1]][n-2^Floor[Log[2, n]] ]];
a[n_] := a[n] = If[n == 0, 0, a[n-1] + 2*b[n-1] + b[n]];
Array[a, 100] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Clarified definition by Omar E. Pol, Feb 06 2017

cp's OEIS Frontend

A151574 a(0)=1, a(1)=2; a(2^i + j) = a(j) + 2*a(j+1) for 0 <= j < 2^i.

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A151704 a(0)=1, a(1)=0; a(2^i+j) = 2*a(j) + a(j+1) for 0 <= j < 2^i.

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A151705 a(0)=0, a(1)=1; a(2^i+j) = 2*a(j) + 2*a(j+1) for 0 <= j < 2^i.

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A151706 a(0)=1, a(1)=0; a(2^i+j) = 2*a(j) + 2*a(j+1) for 0 <= j < 2^i.

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A151707 a(0)=1, a(1)=1; a(2^i+j) = 2*a(j) + 2*a(j+1) for 0 <= j < 2^i.

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A151708 a(0)=1, a(1)=2; a(2^i+j)=2*a(j)+2*a(j+1) for 0 <= j < 2^i.

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A169708 First differences of A169707.

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A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

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A162956 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.

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A151705 a(0)=0, a(1)=1; a(2^i+j) = 2a(j) + 2a(j+1) for 0 <= j < 2^i.

A151706 a(0)=1, a(1)=0; a(2^i+j) = 2a(j) + 2a(j+1) for 0 <= j < 2^i.

A151707 a(0)=1, a(1)=1; a(2^i+j) = 2a(j) + 2a(j+1) for 0 <= j < 2^i.

A151708 a(0)=1, a(1)=2; a(2^i+j)=2a(j)+2a(j+1) for 0 <= j < 2^i.