A170886 -id:A170886 - cp's OEIS frontend

A170887 First differences of toothpick sequence A170886.

0, 1, 2, 2, 2, 4, 6, 6, 6, 8, 12, 6, 8, 12, 18, 14, 14, 20, 20, 6, 8, 12, 18, 14, 16, 24, 26, 16, 24, 38, 46, 38, 42, 52, 36, 6, 8, 12, 18, 14, 16, 24, 26, 16, 24, 38, 46, 38, 44, 56, 42, 16, 24, 38, 46, 40, 52, 70, 64, 52, 82, 118, 126, 114, 130, 132, 68, 6, 8, 12, 18, 14, 16, 24

Offset: 0

Omar E. Pol, Jan 09 2010

Number of toothpicks added at n-th stage to the toothpick structure of A170886. - Omar E. Pol, Jan 31 2013

			From _Omar E. Pol_, Jan 30 2013: (Start)
If written as an irregular triangle in which rows 0..3 have length 1, it appears that row j has length 2^(j-4), if j >= 4. - _Omar E. Pol_, Jan 31 2013
0;
1;
2;
2;
2;
4,6;
6,6,8,12;
6,8,12,18,14,14,20,20;
6,8,12,18,14,16,24,26,16,24,38,46,38,42,52,36;
6,8,12,18,14,16,24,26,16,24,38,46,38,44,56,42,16,24,38,46,40,52,70,64,52,82,118,126,114,130,132,68;
6,8,12,18,14,16,24,...
(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

Cf. A139250, A139251, A160407, A170886, A170889, A170891, A170893.

More terms from R. J. Mathar, Jan 25 2010

A160406 Toothpick sequence starting at the vertex of an infinite 90-degree wedge.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 14, 18, 20, 22, 26, 30, 34, 40, 50, 58, 60, 62, 66, 70, 74, 80, 90, 98, 102, 108, 118, 128, 140, 160, 186, 202, 204, 206, 210, 214, 218, 224, 234, 242, 246, 252, 262, 272, 284, 304, 330, 346, 350, 356, 366, 376, 388, 408, 434, 452, 464, 484, 512, 542, 584

Offset: 0

Omar E. Pol, May 23 2009

nonn

Consider the wedge of the plane defined by points (x,y) with y >= |x|, with the initial toothpick extending from (0,0) to (0,2); then extend by the same rule as for A139250, always staying inside the wedge.
Number of toothpick in the structure after n rounds.
The toothpick sequence A139250 is the main entry for this sequence. See also A153000. First differences: A160407.

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.]
Omar E. Pol, Illustration of initial terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A153000, A153006, A152980, A160407, A160408, A160409.

Cf. A170886-A170895.

G := (x + 2*x^2 + 4*x^2*(1+x)*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..20)-1)/(1+2*x))/(1-x); P:=(G + 2 + x*(5-x)/(1-x)^2)*x/(2*(1+x)); series(P,x,200); seriestolist(%); # N. J. A. Sloane, May 25 2009

terms = 62;
G = (x + 2x^2 + 4x^2 (1+x)(Product[1+x^(2^k-1) + 2x^(2^k), {k, 1, Ceiling[ Log[2, terms]]}]-1)/(1+2x))/(1-x);
P = (G + 2 + x(5-x)/(1-x)^2) x/(2(1+x));
CoefficientList[P + O[x]^terms, x] (* Jean-François Alcover, Nov 03 2018, from Maple *)

A139250(n) = 2a(n) + 2a(n+1) - 4n - 1 for n > 0. - N. J. A. Sloane, May 25 2009

Let G = (x + 2*x^2 + 4*x^2*(1+x)*((Product_{k>=1} (1 + x^(2^k-1) + 2*x^(2^k))) - 1)/(1+2*x))/(1-x) (= g.f. for A139250); then the g.f. for the present sequence is (G + 2 + x*(5-x)/(1-x)^2)*x/(2*(1+x)). - N. J. A. Sloane, May 25 2009

More terms from N. J. A. Sloane, May 25 2009

Definition revised by N. J. A. Sloane, Jan 02 2010

A170888 Similar to A160406, but always staying outside the wedge, starting at stage 0 with a vertical half-toothpick which protrudes from the vertex of the wedge.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 59, 69, 81, 101, 127, 143, 147, 153, 163, 173, 185, 205, 231, 249, 261, 281, 309, 339, 381, 445, 511, 543, 547, 553, 563, 573, 585, 605, 631, 649, 661, 681, 709, 739, 781, 845, 911, 945, 957, 977, 1005, 1035, 1077, 1141

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

See A170889 for the first differences.

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, A170886, A170889, A170890, A170892.

Terms beyond a(10) from R. J. Mathar, Jan 25 2010

A170890 Toothpick sequence similar to A160406, but always staying outside the wedge, starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 14, 21, 29, 37, 43, 53, 61, 71, 83, 103, 123, 139, 151, 165, 173, 183, 195, 215, 235, 253, 271, 295, 317, 345, 385, 441, 493, 531, 559, 581, 589, 599, 611, 631, 651, 669, 687, 711, 733, 761, 801, 857, 909, 949, 983, 1015, 1037, 1065, 1105, 1161

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

The initial half-tookpick makes an angle of 90 degrees w.r.t. the wedge's direction. This breaks the symmetry and explains the changing parity of the terms. - M. F. Hasler, Jan 29 2013

			From _M. F. Hasler_, Jan 29 2013: (Start)
The first steps are illustrated as follows, where two vertical "|" or three horizontal "_" correspond to one single full toothpick:
:                                ___ ___  |___ ___|
:                 ___    |___|    |___|   | |___| |
:   _      |_      |_    | |_|    | |_|   | | |_|
:   /\     |/\     |/\     |/\   ¯¯¯|/\   |¯¯¯|/\
:  /  \    /  \    /  \    /  \     /  \      /  \
:
: a(0)=0, a(1)=1, a(2)=2, a(3)=4, a(5)=7, a(6)=10, ... (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

See A170891 for the first differences.

Cf. A139250, A170886, A170888, A170892.

A170890(n, print_all=0)={ my( cnt=n>0, ee=[[1,1]], p=Set(vector(2*n-cnt,k,k-n-abs(k-n)*I)), c, d); for(i=2, n, print_all & print1(cnt","); p=setunion(p, Set(Mat(ee~)[, 1])); my(ne=[]); for(k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); cnt+=#ee); cnt} \\ - M. F. Hasler, Jan 29 2013

a(9) corrected by Omar E. Pol, following an observation by Kevin Ryde, Jan 29 2013

Terms beyond a(9) from M. F. Hasler, Jan 29 2013

A170892 Toothpick sequence similar to A160406, but always staying outside the wedge, starting at stage 1 with a vertical toothpick whose endpoint touches the vertex of the wedge.

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 16, 24, 34, 44, 48, 56, 66, 78, 90, 112, 138, 156, 160, 168, 178, 190, 202, 224, 250, 270, 282, 304, 332, 364, 406, 472, 538, 572, 576, 584, 594, 606, 618, 640, 666, 686, 698, 720, 748, 780, 822, 888, 954, 990, 1002, 1024, 1052, 1084, 1126, 1192, 1260, 1308, 1350, 1418, 1502, 1604, 1750, 1944

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

See A170893 for the first differences.

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, A160406, A170886, A170888, A170890, A170893.

A170892(n, print_all=0)={my( ee=[[2*I, I]], p=Set( concat( vector( 2*n-(n>0),k,k-n-abs(k-n)*I ), I )), cnt=2); print_all & print1("1,2"); n<3 & return(n); for(i=3, n, p=setunion(p, Set(Mat(ee~)[, 1])); my(c, d, ne=[]); for( k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); cnt+=#ee; print_all & print1(","cnt)); cnt} \\ - M. F. Hasler, Jan 30 2013

Terms beyond a(10) from M. F. Hasler, Jan 30 2013

cp's OEIS Frontend

A170887 First differences of toothpick sequence A170886.

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A160406 Toothpick sequence starting at the vertex of an infinite 90-degree wedge.

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A170888 Similar to A160406, but always staying outside the wedge, starting at stage 0 with a vertical half-toothpick which protrudes from the vertex of the wedge.

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Author

Keywords

Comments

Links

Crossrefs

Extensions

A170890 Toothpick sequence similar to A160406, but always staying outside the wedge, starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A170892 Toothpick sequence similar to A160406, but always staying outside the wedge, starting at stage 1 with a vertical toothpick whose endpoint touches the vertex of the wedge.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions