A151550 -id:A151550 - cp's OEIS frontend

A151553 G.f.: (1 + x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

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, 7, 8, 9, 12, 14, 11, 6, 5, 7, 8, 9, 12, 14, 12, 10, 12, 15, 17, 21, 26, 25, 16, 6, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 6, 5, 7, 8, 9, 12, 14, 12, 10, 12, 15, 17, 21, 26

Offset: 0

N. J. A. Sloane, May 20 2009

			If formatted as a triangle:
.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,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,6,2,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,7
... 5,7,8,9,12,14,12,10,12,15,17,21,26,25,17,11,12,15,17,21,26,26,22,22,27,32,38,47,51,41,22,7,2,
.3,4,4,5,7,7,4,2, ...

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.

Cf. A139250, A151550, A151551, A151552.

CoefficientList[Series[(1+x)Product[1+x^(2^n-1)+x^2^n,{n,10}],{x,0,100}],x] (* Harvey P. Dale, Jul 13 2019 *)
a[n_] := Sum[If[OddQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, 2*Floor[Log2[n + 1]] + 1}]; Array[a, 92, 0] (* Amiram Eldar, Jul 29 2023 *)

Recurrence: a(0)=1, a(1) = a(2) = 2; a(2^m-1)=2 for m >= 2; a(2^m) = 3 for m >= 2; a(2^m-2) = m for m >= 3; otherwise, for m >= 5, if m=2^i+j (0 <= j < 2^i - 1), a(m) = a(j) + a(j+1).

a(n) = Sum_{k>=0, n+k odd} binomial(A000120(n+k),k); the sum may be restricted further to k <= 2*A000523(n+1)+1 [based on Hagen von Eitzen's formula for A151552]. [corrected by Amiram Eldar, Jul 29 2023]

A151551 G.f.: (1 + 3x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

1, 4, 5, 7, 6, 13, 16, 13, 6, 13, 17, 20, 25, 42, 44, 25, 6, 13, 17, 20, 25, 42, 45, 32, 25, 43, 54, 65, 92, 128, 112, 49, 6, 13, 17, 20, 25, 42, 45, 32, 25, 43, 54, 65, 92, 128, 113, 56, 25, 43, 54, 65, 92, 129, 122, 89, 93, 140, 173, 222, 312, 368, 272, 97, 6, 13, 17, 20, 25, 42

Offset: 0

N. J. A. Sloane, May 19 2009

nonn

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, A151550, A151552.

A151555 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

Original entry on oeis.org

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, 33, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 81, 39, 20, 33, 41, 50, 72, 97, 89, 66, 73, 107, 132, 172, 240, 272, 192, 65, 5, 10, 13, 15, 20, 32, 33, 23

Offset: 0

N. J. A. Sloane, May 20 2009

nonn

From Gary W. Adamson, May 25 2009: (Start)
Convolved with A078008 signed (A151575) [1, 0, 2, -2, 6, -10, 22, -42, 86, -170, ...]
equals the toothpick sequence A153006: (1, 3, 6, 9, 13, 20, 28, ...). (End)
If A151550 is written as a triangle then the rows converge to this sequence. - N. J. A. Sloane, Jun 16 2009

			From _Omar E. Pol_, Jun 19 2009: (Start)
May be written as a triangle:
1;
3;
4,5;
5,10,12,9;
5,10,13,15,20,32,32,17;
5,10,13,15,20,32,33,23,20,33,41,50,72,96,80,33;
5,10,13,15,20,32,33,23,20,33,41,50,72,96,81,39,20,33,41,50,72,97,89,66,73,...
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..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, A151551, A151552, A151553, A151554, A151550, A152980, A153006.

Cf. A078008. - Gary W. Adamson, May 25 2009

A151554 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

Original entry on oeis.org

1, 3, 3, 3, 4, 6, 5, 3, 4, 6, 6, 7, 10, 11, 7, 3, 4, 6, 6, 7, 10, 11, 8, 7, 10, 12, 13, 17, 21, 18, 9, 3, 4, 6, 6, 7, 10, 11, 8, 7, 10, 12, 13, 17, 21, 18, 10, 7, 10, 12, 13, 17, 21, 19, 15, 17, 22, 25, 30, 38, 39, 27, 11, 3, 4, 6, 6, 7, 10, 11, 8, 7, 10, 12, 13, 17, 21, 18, 10, 7, 10, 12, 13

Offset: 0

N. J. A. Sloane, May 20 2009

nonn

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

Cf. A139250, A151550, A151551, A151552, A151553.

A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 4, 2, 4, 6, 6, 8, 14, 15, 8, 2, 4, 6, 6, 8, 14, 16, 10, 8, 14, 18, 20, 30, 44, 39, 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, 95

Offset: 1

Omar E. Pol, Mar 22 2011, Apr 06 2011

nonn

It appears that both a(2) and a(2^k - 1) are odd numbers, for k >= 2. Other terms are even numbers.

			At stage 1 we start in the first quadrant from a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1). There are no gulls in the structure, so a(1) = 0.
At stage 2 we place a gull (or G-toothpick) with its midpoint at (1,1) and its endpoints at (2,0) and (2,2), so a(2) = 1. There is only one exposed midpoint at (2,2).
At stage 3 we place a gull with its midpoint at (2,2), so a(3) = 1. There are two exposed endpoints.
At stage 4 we place two gulls, so a(4) = 2. There are two exposed endpoints.
At stage 5 we place two gulls, so a(5) = 2. There are four exposed endpoints.
And so on.
If written as a triangle begins:
0,
1,
1,2,
2,4,5,4,
2,4,6,6,8,14,15,8,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,39,16,
2,4,6,6,8,14,16,10,8,14,18,20,30,44,40,18,8,14,18,20,30,44,42,28,...
It appears that rows converge to A151688.

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. A099035, A151550, A151688, A187210, A187211, A187212, A187213, A187220.

read("transforms3") ;
L := BFILETOLIST("b187212.txt") ;
A187213 := DIFF(L) ;
seq( op(n,A187213)/2,n=2..nops(A187213)) ; # R. J. Mathar, Mar 30 2011

a(1)=0. a(n) = A187213(n)/2, for n >= 2.

It appears that a(2^k - 1) = A099035(k-1), for k >= 2.

cp's OEIS Frontend

A151553 G.f.: (1 + x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

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A151551 G.f.: (1 + 3x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

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A151555 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + 2*x^(2^n)).

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A151554 G.f.: (1 + 2x) * Product_{n>=1} (1 + x^(2^n-1) + x^(2^n)).

Views

Author

Keywords

Links

Crossrefs

A187214 Number of gulls (or G-toothpicks) added at n-th stage in the first quadrant of the gullwing structure of A187212.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula