A170885 -id:A170885 - cp's OEIS frontend

A160160 Toothpick sequence in the three-dimensional grid.

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

A170884 In the toothpick structure of A160160, the number of nodes occupied after n steps, assuming that the toothpicks have length 2.

Original entry on oeis.org

0, 3, 7, 15, 27, 39, 51, 67, 99, 155, 223, 263, 283, 299, 331, 387, 467, 555, 659, 811, 1067, 1443, 1831, 1995, 2059, 2083, 2123, 2179, 2259, 2347, 2451, 2603, 2859, 3235, 3659, 3955, 4211, 4483, 4899, 5451

Offset: 0

David Applegate, Omar E. Pol and N. J. A. Sloane, Jan 09 2010

nonn

See A170885 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, A170885.

a(7)-a(39) from Nathaniel Johnston, Nov 13 2010

A170891 First differences of the toothpick sequence A170890.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 7, 8, 8, 6, 10, 8, 10, 12, 20, 20, 16, 12, 14, 8, 10, 12, 20, 20, 18, 18, 24, 22, 28, 40, 56, 52, 38, 28, 22, 8, 10, 12, 20, 20, 18, 18, 24, 22, 28, 40, 56, 52, 40, 34, 32, 22, 28, 40, 56, 54, 50, 56, 66, 68, 92, 132, 160, 138, 98, 68, 38

Offset: 0

Omar E. Pol, Jan 09 2010

Number of toothpicks added at n-th stage to the toothpick structure of A170890. - Omar E. Pol, Jan 31 2013

			From _Omar E. Pol_, Jan 31 2013 (Start):
If written as an irregular triangle in which rows 0..4 have length 1, it appears that row j has length 2^(j-5), if j >= 5.
0;
1;
1;
2;
3;
3;
4,7;
8,8,6,10;
8,10,12,20,20,16,12,14;
8,10,12,20,20,18,18,24,22,28,40,56,52,38,28,22;
8,10,12,20,20,18,18,24,22,28,40,56,52,40,34,32,22,28,40,56,54,50,56,66,68,92,132,160,138,98,68,38;
(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, A170890, A170885, A170887, A170889, A170893.

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

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A160160 Toothpick sequence in the three-dimensional grid.

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A170884 In the toothpick structure of A160160, the number of nodes occupied after n steps, assuming that the toothpicks have length 2.

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A170891 First differences of the toothpick sequence A170890.

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