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
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
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
Comments