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