A152980 -id:A152980 - cp's OEIS frontend

A151552 G.f.: Product_{k>=1} (1 + x^(2^k-1) + x^(2^k)).

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 4, 3, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 2, 3, 4, 4, 5, 7, 7, 5, 5

Offset: 0

N. J. A. Sloane, May 19 2009, Dec 26 2009

			Written as a triangle:
1;
1;
1,1;
2,2,1,1;
2,2,2,3,4,3,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,3,4,4,5,7,7,5,5,...
The rows converge to A151714.

N. J. A. Sloane, 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.

For generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.

Cf. A139250, A151550, A151551, A160573, A151702, A151714, A151553.

G := mul( 1 + x^(2^n-1) + x^(2^n), n=1..20);
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=proc(n) local t1,k; global wt; t1:=0; for k from 0 to 20 do if n+k mod 2 = 0 then t1:=t1+binomial(wt(n+k),k); fi; od; t1; end;

a[n_] := Sum[If[EvenQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, Floor[Log2[n + 1]]}]; Array[a, 100, 0] (* Amiram Eldar, Jul 29 2023 *)

a(n) = 1 for 0 <= n <= 3; thereafter write n = 2^i + j, with 0 <= j < 2^i, then a(n) = a(j) + a(j+1), except that a(2^(i+1)-2) = a(2^(i+1)-1) = 1.

a(n) = Sum_{k>=0, n+k even} binomial(A000120(n+k),k); the sum may be restricted further to k <= A000523(n+1). - Hagen von Eitzen, May 20 2009 [corrected by Amiram Eldar, Jul 29 2023]

A152968 a(n) = A139251(n+1)/2.

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 4, 2, 4, 6, 6, 8, 14, 16, 8, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 16, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 32, 2

Offset: 1

Omar E. Pol, Dec 16 2008, Dec 20 2008

nonn

Also, first differences of toothpicks numbers A152998. [From Omar E. Pol, Jan 02 2009]

			Triangle begins:
.1;
.2,2;
.2,4,6,4;
.2,4,6,6,8,14,16,8;
.2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,16;
....
Rows approach A151688. - _N. J. A. Sloane_, Jun 03 2009

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, A139251, A139252, A152978, A152980, A153006, A152998.

Write n = 2^i +j, 0 <= j < 2^i; then a(n) = Sum_k 2^(wt(j+k)-k)*binomial(wt(j+k),k). except that a(2^r-1) = 2^(r-1). - N. J. A. Sloane, Jun 03 2009, Jul 16 2009

G.f.: x*(Prod(1+x^(2^k-1)+2*x^(2^k),k=0..oo)-1)/(1+2*x). - N. J. A. Sloane, Jun 05 2009

A151550 Expansion of g.f. Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 4, 1, 3, 4, 5, 5, 10, 12, 8, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 16, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 80, 32, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13

Offset: 0

N. J. A. Sloane, May 19 2009, Jun 17 2009

nonn

When convolved with [1, 2, 2, 2, ...] gives the toothpick sequence A153006: (1, 3, 6, 9, ...). - Gary W. Adamson, May 25 2009

This sequence and the Adamson's comment both are mentioned in the Applegate-Pol-Sloane article, see chapter 8 "generating functions". - Omar E. Pol, Sep 20 2011

			From _Omar E. Pol_, Jun 09 2009, edited by _N. J. A. Sloane_, Jun 17 2009:
May be written as a triangle:
  0;
  1;
  1,2;
  1,3,4,4;
  1,3,4,5,5,10,12,8;
  1,3,4,5,5,10,12,9,5,10,13,15,20,32,32,16;
  1,3,4,5,5,10,12,9,5,10,13,15,20,32,32,17,5,10,13,15,20,32,33,23,20,33,41,...
The rows of the triangle converge to A151555.

D. 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

N. J. A. Sloane, 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.], which is also available at arXiv:1004.3036v2, [math.CO], 2010.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For generating functions of the form Product_{k>=c} (1+a*x^(2^k-1)+b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Cf. A139250, A151551, A151552, A151553, A151554, A151555, A152980, A153006, A151688.
Cf. A000079. - Omar E. Pol, Jun 09 2009

terms = 100;
CoefficientList[Product[(1+x^(2^n-1) + 2 x^(2^n)), {n, 1, Log[2, terms] // Ceiling}] + O[x]^terms, x] (* Jean-François Alcover, Aug 05 2018 *)

To get a nice recurrence, change the offset to 0 and multiply the g.f. by x as in the triangle in the example lines. Then we have: a(0)=0; a(2^i)=1; a(2^i-1)=2^(i-1) for i >= 1; otherwise write n = 2^i+j with 1 <= j <= 2^i-2, then a(n) = a(2^i+j) = 2*a(j) + a(j+1).

A160573 G.f.: Product_{k >= 0} (1 + x^(2^k-1) + x^(2^k)).

Original entry on oeis.org

2, 3, 3, 3, 5, 6, 4, 3, 5, 6, 6, 8, 11, 10, 5, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 6, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 8, 8, 11, 12, 14, 19, 21, 17, 15, 19, 23, 26, 33, 40, 36, 21, 7, 3, 5, 6, 6, 8, 11, 10, 7, 8, 11, 12, 14, 19, 21, 15, 8, 8

Offset: 0

Hagen von Eitzen, May 20 2009

nonn

Sequence mentioned in the Applegate-Pol-Sloane article; see Section 9, "explicit formulas." - Omar E. Pol, Sep 20 2011

			a(5) = binomial(2,0) + binomial(2,1) + binomial(3,2) + binomial(1,3) + binomial(2,4) + binomial(2,5) + ... = 1 + 2 + 3 + 0 + 0 + 0 + ... = 6
From _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
2;
3;3;
3,5,6,4;
3,5,6,6,8,11,10,5;
3,5,6,6,8,11,10,7,8,11,12,14,19,21,15,6;
3,5,6,6,8,11,10,7,8,11,12,14,19,21,15,8,8,11,12,14,19,21,17,15,19,23,26,...
(End)

D. 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, 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

For generating functions of the form Product_{k>=c} (1+a*x^(2^k-1)+b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Row sums of A151683. See A151687 for another version.
Cf. A151552 (g.f. has one factor fewer).
Limiting form of rows of A118977 when that sequence is written as a triangle and the initial 1 is omitted. - N. J. A. Sloane, Jun 01 2009
Cf. A139250, A139251. - Omar E. Pol, Sep 20 2011

Maple
```
See A118977 for Maple code.
```

max = 80; Product[1 + x^(2^k - 1) + x^(2^k), {k, 0, Ceiling[Log[2, max]]}] + O[x]^max // CoefficientList[#, x]& (* Jean-François Alcover, Nov 10 2016 *)

a(n) = Sum_{i >= 0} binomial(A000120(n+i),i).
For k >= 1, a(2^k-2) = k+1 and a(2^k-1) = 3; otherwise if n = 2^i + j, 0 <= j <= 2^i-3, a(n) = a(j) + a(j+1).
a(n) = 2*A151552(n) + A151552(n-1).

A151688 G.f.: Product_{n>=0} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 40, 18, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 34, 8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 98, 44, 30, 46, 56, 70, 104, 130, 112, 86, 106, 148, 182, 244, 336, 352, 224, 66, 8, 14, 18, 20, 30, 44

Offset: 0

Omar E. Pol, May 02 2009

This is essentially the same g.f. as A151550 but with the n=0 term included.

			If written as a triangle, begins:
  2;
  4;
  6, 6;
  8, 14, 16, 10;
  8, 14, 18, 20, 30, 44, 40, 18;
  8, 14, 18, 20, 30, 44, 42, 28, 30, 46, 56, 70, 104, 128, 96, 34;
  ...

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

Equals 2*A152980 = A147646/2.
Equals limit of rows of triangle in A152968.
For generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Cf. A151550, A139251, A139250.

terms = 70; CoefficientList[Product[(1+x^(2^n-1) + 2 x^(2^n)), {n, 0, Log[2, terms] // Ceiling}] + O[x]^terms, x] (* Stefano Spezia, Sep 02 2025 *)

a(n) = Sum_{k>=0} 2^(wt(n+k)-k)*binomial(wt(n+k),k).

Edited by N. J. A. Sloane, Jun 03 2009, Jul 14 2009

A147646 If A139251 is written as a triangle with rows of lengths 1, 2, 4, 8, 16, ..., the n-th row begins with 2^n followed by the first 2^n-1 terms of the present sequence.

Original entry on oeis.org

4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 36, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 192, 68, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 196, 88, 60, 92, 112, 140, 208, 260, 224, 172, 212, 296, 364, 488, 672, 704, 448, 132

Offset: 1

David Applegate, Apr 30 2009

nonn

Limiting behavior of the rows of the triangle in A139251 when the first column of that triangle is omitted.
First differences of A159795. - Omar E. Pol, Jul 24 2009
It appears that a(n) is also the number of new grid points that are covered at n-th stage of A139250 version "Tree", assuming the toothpicks have length 4, 3, and 2 (see also A159795 and A153006). - Omar E. Pol, Oct 25 2011

			Further comments: A139251 as a triangle is:
. 1
. 2 4
. 4 4 8 12
. 8 4 8 12 12 16 28 32
. 16 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80
. 32 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...
leading to the present sequence:
. 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...
Note that this can also be written as a triangle:
. 4 8
. 12 12 16 28
. 32 20 16 28 36 40 60 88
. 80 36 16 28 36 40 60 88 84 56 60 92 112 140 208 256
. 192 68 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
The first column is (n+1)2^n (where n is the row number),
the second column is 2^(n+1)+4,
and the rest exhibits the same constant column behavior,
where the rows converge to:
. 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
Once again this can be written as a triangle:
. 16
. 28 36 40 60
. 88 84 56 60 92 112 140 208
. 256 196 88 60 92 112 140 208 260 224 172 212 296 364 488 672
. 704 452 152 60 92 112 140 208 260 224 172 212 296 364 488 672 708 480 236 ...
and this behavior continues ad infinitum.

David Applegate, Table of n, a(n) for n = 1..2047
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
Index entries for sequences related to toothpick sequences

Equals 2*A151688 and 4*A152980. - N. J. A. Sloane, Jul 16 2009

Cf. A139250, A139251, A153006, A159795, A151548.

S:=proc(n) option remember; local i,j;
if n <= 0 then RETURN(0); fi;
if n <= 2 then RETURN(2^(n+1)); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then RETURN(2*n+4); fi;
if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i-1 then RETURN(2*S(j)+S(j+1)-4); fi;
-1;
end; # N. J. A. Sloane, May 18 2009

Letting n = 2^i + j for 0 <= j < 2^i, we have the recurrence (see A139251 for proof):
a(1) = 4
a(2) = 8
a(n) = 2n+4 = 2*a(n/2) - 4 if j = 0
a(n) = 2*a(j) + a(j+1) - 4 if j = 2^i-1
a(n) = 2*a(j) + a(j+1) if 1 <= j < 2^i-1
It appears that a(n) = A151548(n-1) + A151548(n). - Omar E. Pol, Feb 19 2015

A151685 a(n) = Sum_{k >= 0} bin2(wt(n+k),k+1), where bin2(i,j) = A013609(i,j), wt(i) = A000120(i).

Original entry on oeis.org

3, 7, 5, 7, 17, 17, 7, 7, 17, 17, 19, 41, 51, 31, 9, 7, 17, 17, 19, 41, 51, 31, 21, 41, 51, 55, 101, 143, 113, 49, 11, 7, 17, 17, 19, 41, 51, 31, 21, 41, 51, 55, 101, 143, 113, 49, 23, 41, 51, 55, 101, 143, 113, 73, 103, 143, 161, 257, 387, 369, 211, 71, 13, 7, 17, 17, 19, 41, 51

Offset: 0

N. J. A. Sloane, Jun 01 2009

nonn

Or, a(n) = Sum_{k >= 0} 2^wt(k) * binomial(wt(n+k),k).

			Contribution from _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
.3;
.7,5;
.7,17,17,7;
.7,17,17,19,41,51,31,9;
.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,11;
.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,23,41,51,55,101,143,113,...
(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. [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 generating functions of the form Product_{k>=c} (1+a*x^(2^k-1)+b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Cf. A151689, A151691.
Cf. A000079. - Omar E. Pol, Jun 09 2009

bin2:=proc(n,k) option remember; if k<0 or k>n then 0
elif k=0 then 1 else 2*bin2(n-1,k-1)+bin2(n-1,k); fi; end;
wt := proc(n) local w,m,i;
w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=n->add( bin2(wt(n+k),k),k=0..120 );
# or:
f := n->add( 2^k*binomial(wt(n+k),k),k=0..20 );

max = 70; (* number of terms *)
CoefficientList[Product[1 + 2*x^(2^k-1) + x^(2^k), {k, 0, Log2[max+1] // Ceiling}] + O[x]^max, x] (* Jean-François Alcover, Aug 03 2022 *)

G.f.: Product_{ k >= 0 } (1 + 2*x^(2^k-1) + x^(2^k)).

A151691 G.f.: Product_{k>=1} (1 + 2*x^(2^k-1) + x^(2^k)).

Original entry on oeis.org

1, 2, 1, 2, 5, 4, 1, 2, 5, 4, 5, 12, 13, 6, 1, 2, 5, 4, 5, 12, 13, 6, 5, 12, 13, 14, 29, 38, 25, 8, 1, 2, 5, 4, 5, 12, 13, 6, 5, 12, 13, 14, 29, 38, 25, 8, 5, 12, 13, 14, 29, 38, 25, 16, 29, 38, 41, 72, 105, 88, 41, 10, 1, 2, 5, 4, 5, 12, 13, 6, 5, 12, 13, 14, 29, 38, 25, 8, 5, 12, 13, 14, 29

Offset: 0

N. J. A. Sloane, Jun 04 2009

nonn

			From _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
  1;
  2,1;
  2,5,4,1;
  2,5,4,5,12,13,6,1;
  2,5,4,5,12,13,6,5,12,13,14,29,38,25,8,1;
  2,5,4,5,12,13,6,5,12,13,14,29,38,25,8,5,12,13,14,29,38,25,16,29,38,41,72,...
(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. [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 generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Cf. A151685. See A151703 for another version with a simpler recurrence.
Cf. A000079. - Omar E. Pol, Jun 09 2009

A151692 G.f.: Product_{k>=2} (1 + x^(2^k-1) + x^(2^k)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 1, 1, 3, 3, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 1, 1, 3, 3, 1, 0, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 2, 4, 6, 4, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 1, 1, 3, 3, 1, 0, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 2, 4, 6, 4, 1, 0, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 2

Offset: 0

N. J. A. Sloane, Jun 04 2009

nonn

			From _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
  1;
  0,0;
  1,1,0,0;
  1,1,0,1,2,1,0,0;
  1,1,0,1,2,1,0,1,2,1,1,3,3,1,0,0;
  1,1,0,1,2,1,0,1,2,1,1,3,3,1,0,1,2,1,1,3,3,1,1,3,3,2,4,6,4,1,0,0;
  1,1,0,1,2,1,0,1,2,1,1,3,3,1,0,1,2,1,1,3,3,1,1,3,3,2,4,6,4,1,0,1,2,1,1,3,3,...
(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. [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 generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Cf. A160573, A151552.
Cf. A000079. - Omar E. Pol, Jun 09 2009

A151693 G.f.: Product_{k>=0} (1 + 2x^(2^k-1) + 2x^(2^k)).

Original entry on oeis.org

3, 8, 10, 10, 22, 36, 28, 14, 22, 36, 40, 64, 116, 128, 72, 22, 22, 36, 40, 64, 116, 128, 84, 72, 116, 152, 208, 360, 488, 400, 176, 38, 22, 36, 40, 64, 116, 128, 84, 72, 116, 152, 208, 360, 488, 400, 188, 88, 116, 152, 208, 360, 488, 424, 312, 376, 536, 720, 1136, 1696, 1776

Offset: 0

N. J. A. Sloane, Jun 04 2009

nonn

			From _Omar E. Pol_, Jun 09 2009: (Start)
Triangle begins:
   3;
   8,10;
  10,22,36,28;
  14,22,36,40,64,116,128,72;
  22,22,36,40,64,116,128,84,72,116,152,208,360,488,400,176;
  38,22,36,40,64,116,128,84,72,116,152,208,360,488,400,188,88,116,152,208,...
(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. [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 generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.

Cf. A000079. - Omar E. Pol, Jun 09 2009

cp's OEIS Frontend

A151552 G.f.: Product_{k>=1} (1 + x^(2^k-1) + x^(2^k)).

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A152968 a(n) = A139251(n+1)/2.

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A151550 Expansion of g.f. Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)).

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A160573 G.f.: Product_{k >= 0} (1 + x^(2^k-1) + x^(2^k)).

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A151688 G.f.: Product_{n>=0} (1 + x^(2^n-1) + 2*x^(2^n)).

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A147646 If A139251 is written as a triangle with rows of lengths 1, 2, 4, 8, 16, ..., the n-th row begins with 2^n followed by the first 2^n-1 terms of the present sequence.

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A151685 a(n) = Sum_{k >= 0} bin2(wt(n+k),k+1), where bin2(i,j) = A013609(i,j), wt(i) = A000120(i).

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A151693 G.f.: Product_{k>=0} (1 + 2x^(2^k-1) + 2x^(2^k)).