login
A160161
First differences of the 3D toothpick numbers A160160.
13
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, 24, 40, 56, 56, 64, 80, 152, 232, 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
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
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.]
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)=(n=A160160_vec(n))-concat(0, n[^-1]) \\ M. F. Hasler, Dec 11 2018
(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
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