A170876 -id:A170876 - cp's OEIS frontend

A170875 First differences of A170876.

0, 1, 4, 16, 16, 16, 64, 80, 64, 144, 160, 224, 176, 256, 320, 400, 512, 480, 688, 768, 704, 816, 896, 1120, 1168, 1536, 1568, 1936, 1600, 1984, 1776, 2112, 2304, 2544, 2656, 3008, 2896, 3072, 3200

Offset: 0

N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar, Jun 02 2009; revised by R. J. Mathar, Jan 08 2010

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. A170876.

More terms from Colin Barker, Apr 19 2015

A160160 Toothpick sequence in the three-dimensional grid.

Original entry on oeis.org

0, 1, 3, 7, 15, 23, 31, 39, 55, 87, 143, 175, 191, 199, 215, 247, 303, 359, 423, 503, 655, 887, 1239, 1383, 1431, 1463, 1487, 1527, 1583, 1639, 1703, 1783, 1935, 2167, 2519, 2735, 2903, 3079, 3351, 3711, 4207, 4655, 5191, 5855, 7023, 8511, 10511, 11279, 11583, 11919, 12183, 12375, 12487, 12607

Offset: 0

Omar E. Pol, May 03 2009, May 06 2009

Similar to A139250, except the toothpicks are placed in three dimensions, not two. The first toothpick is in the z direction. Thereafter, new toothpicks are placed at free ends, as in A139250, perpendicular to the existing toothpick, but choosing in rotation the x-direction, y-direction, z-direction, x-direction, etc.

The graph of this sequence has a nice self-similar shape: it looks the when the x-range is multiplied by 2, e.g. a(0..125) vs a(0..250) or a(0..500). - M. F. Hasler, Dec 12 2018

M. F. Hasler, Table of n, a(n) for n = 0..500
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.]
R. J. Mathar, C++ program
R. J. Mathar, View after stage 1
R. J. Mathar, View after stage 2
R. J. Mathar, View after stage 3
R. J. Mathar, View after stage 4
R. J. Mathar, View after stage 5
R. J. Mathar, View after stage 6
R. J. Mathar, View after stage 7
R. J. Mathar, View after stage 8
R. J. Mathar, View after stage 9
R. J. Mathar, View after stage 10
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Alex van den Brandhof and Paul Levrie, Tandenstokerrij, Pythagoras, Viskundetijdschrift voor Jongeren, 55ste Jaargang, Nummer 6, Juni 2016, (see page 19 and the back cover).

Cf. A139250, A160120, A160161, A160170, A170884, A170885, A170876.

A160160_vec(n,o=1)={local(s(U)=[Vecsmall(Vec(V)+U)|V<-E], E=[Vecsmall([1,1,1])], J=[], M,A,B,U); [if(i>4, M+=8*#E=setminus(setunion(A=s(U=matid(3)[i%3+1,]), B=select(vecmin,s(-U))), J=setunion(setunion(setintersect(A,B),E),J)),M=1<M. F. Hasler, Dec 11 2018

A160160(n)=sum(k=1,n,A160161[k]) \\ if A160161=A160161_vec(n) has already been computed. - M. F. Hasler, Dec 12 2018

Partial sums of A160161: a(n) = Sum_{1 <= k <= n} A160161(k) for all n >= 0. - M. F. Hasler, Dec 12 2018

Edited by N. J. A. Sloane, Jan 02 2009
Extended to a(76) with C++ program and illustrations by R. J. Mathar, Jan 09 2010
Extended to 500 terms by M. F. Hasler, Dec 12 2018

A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 13, 21, 45, 77, 109, 165, 245, 325, 413, 525, 685, 853, 1093, 1317, 1661, 1981, 2301, 2645, 3093, 3621, 4157, 4861, 5565, 6437, 7173, 8053, 8893, 9917, 11005, 12261, 13589, 14981, 16397, 17837, 19341, 20997

Offset: 0

Omar E. Pol, May 03 2009

nonn

Here a "X-toothpick" is defined to be a cross with 4 endpoints and a midpoint. Also, a X-toothpick can be represented by set of four connected toothpicks forming a cross.
We start at stage 0 on the Z^3 lattice with no X-toothpicks.
At stage 1 place a X-toothpick.
Rule: each exposed endpoint of the X-toothpicks of the old generation must be touched by the midpoint of a X-toothpick of new generation (see illustrations).
The sequence gives the number of X-toothpicks in the three-dimensional structure after n-th stage. A170171 (the first differences) gives the number of X-toothpicks added at n-th stage.
For a similar sequence but starting with a single toothpick see A170876.
For the X-toothpick sequence on Z^2 lattice see A147562, the Ulam-Warburton cellular automaton.
For more information about the growth of toothpicks see A139250.

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.],
R. J. Mathar, C++ program
R. J. Mathar, Views after stages 1 to 10 (PDF)
Olbaid Fractalium, Toothpick sequence Part 1, (Watch from minute 3:14), Youtube video (2023).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A160120, A160160, A160171, A170876.

Cf. A147562. - Omar E. Pol, Mar 28 2011

C++ program, illustrations and more terms (a(6)-a(38)) based on Email from R. J. Mathar dated Jan 10 2010.

A170837 a(0)=0, a(1)=1 and a(n) = 16n-27 for n >= 2.

Original entry on oeis.org

0, 1, 5, 21, 37, 53, 69, 85, 101, 117, 133, 149, 165, 181, 197, 213, 229, 245, 261, 277, 293, 309, 325, 341, 357, 373, 389, 405, 421, 437, 453, 469, 485, 501, 517, 533, 549, 565, 581, 597, 613, 629, 645, 661, 677, 693, 709, 725, 741, 757, 773, 789, 805

Offset: 0

N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar and Benoit Jubin, Jun 02 2009; revised Jan 09 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A170836 (first differences), A170876.

I:=[0, 1, 5, 21]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..60]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[x*(3*x + 12*x^2 + 1)/(x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{2,-1},{0,1,5,21},60] (* Harvey P. Dale, Oct 09 2017 *)

G.f.: x*(3*x+12*x^2+1)/(x-1)^2.
a(n) = 2*a(n-1) -a(n-2), n>=4.
a(n) = 4*A016813(n-2) + 1, n>=2. - Ivan N. Ianakiev, Jul 20 2013

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A170875 First differences of A170876.

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A160160 Toothpick sequence in the three-dimensional grid.

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A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

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A170837 a(0)=0, a(1)=1 and a(n) = 16n-27 for n >= 2.

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