OFFSET
0,3
COMMENTS
Number of toothpicks added at n-th stage to the three-dimensional toothpick structure of A160160.
The sequence should start with a(1) = 1 = A160160(1) - A160160(0), the initial a(0) = 0 seems purely conventional and not given in terms of A160160. The sequence can be written as a table with rows r >= 0 of length 1, 1, 1, 3, 9, 18, 36, ... = 9*2^(r-4) for row r >= 4. In that case, rows 0..3 are filled with 2^r, and all rows r >= 3 have the form (x_r, y_r, x_r) where x_r and y_r have 3*2^(r-4) elements, all multiples of 8. Moreover, y_r[1] = a(A033484(r-2)) = x_{r+1}[1] = a(A176449(r-3)) is the largest element of row r and thus a record value of the sequence. - M. F. Hasler, Dec 11 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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
FORMULA
a(9*2^k - m) = a(6*2^k - m) for all k >= 0 and 2 <= m <= 3*2^(k-1) + 2. - M. F. Hasler, Dec 12 2018
EXAMPLE
Array begins:
===================
x y z
===================
0 1
2 4 8
8 8 8
16 32 56
32 16 8
16 32 56
56 64 80
152 232 352
144 48 32
...
From Omar E. Pol, Feb 28 2018: (Start)
Also, starting with 1, the sequence can be written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 3, as shown below:
1, 2, 4;
8, 8, 8;
8, 16, 32, 56, 32, 16;
8, 16, 32, 56, 56, 64, 80, 152, 232, 352, 144, 48;
32, 24, 40, 56, 56, 64, 80, 152, 232, 352, 216, 168, 176, 272, 360, 496, 448, ...
(End)
If one starts rows with a(A176449(k) = 9*2^k-2), they are of the form A_k, B_k, A_k where A_k and B_k have 3*2^k elements and the first element of A_k is the first element of B_{k-1} and the largest of that (previous) row:
k | a(9*2^k-2, ...) = A_k ; B_k ; A_k
---+-------------------------------------
| a( 1 .. 6) = (1, 2, 4, 8, 8, 8) (One might consider a row (8 ; 8 ; 8).)
0 | a( 7, ...) = (8, 16, 32 ; 56, 32, 16 ; 8, 16, 32)
1 | a(16, ...) = (56, 56, 64, 80, 152, 232 ; 352, 144, 48, 32, 24, 40 ;
| 56, 56, 64, 80, 152, 232)
2 | a(34, ...) = (352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488 ;
| 2000, 768, 304, 336, 264, 192, 112, 120, 128, 112, 168, 240 ;
| 352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488)
3 | a(70, ...) = (2000, 984, ... ; 10576, 4304, ... ; 2000, 984, ...)
4 | a(142, ...) = (10576, 5016, ... ; 54328, 24120, ...; 10576, 5016, ...)
etc. - M. F. Hasler, Dec 11 2018
PROG
(PARI) A160161_vec(n)={local(E=[Vecsmall([1, 1, 1])], s(U)=[Vecsmall(Vec(V)+U)|V<-E], J=[], M, A, B, U); [if(i>4, 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)), 2^(i-1))|i<-[1..n]]} \\ Returns the vector a(1..n). (A160160 is actually given as partial sums of this sequence, rather than the converse.) - M. F. Hasler, Dec 12 2018
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, May 03 2009
EXTENSIONS
Extended to 78 terms with C++ program by R. J. Mathar, Jan 09 2010
Edited and extended by M. F. Hasler, Dec 11 2018
STATUS
approved