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

Author

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

Keywords

• nonn
• look

Comments

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

Links

• 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).

Crossrefs

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

Programs

PARI

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

PARI

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

Formula

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

Extensions

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