A160161 First differences of the 3D toothpick numbers A160160.

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

Author

Omar E. Pol, May 03 2009

Keywords

• nonn
• tabf

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

Examples

			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
		

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
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A160160, A139250, A139251, A160120, A160121, A160171, A170875.

Programs

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

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

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