

A160161


First differences of the 3D toothpick numbers A160160.


11



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
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OFFSET

0,3


COMMENTS

Number of toothpicks added at nth stage to the threedimensional 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^(r4) 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^(r4) elements, all multiples of 8. Moreover, y_r[1] = a(A033484(r2)) = x_{r+1}[1] = a(A176449(r3)) 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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)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


FORMULA

a(9*2^k  m) = a(6*2^k  m) for all k >= 0 and 2 <= m <= 3*2^(k1) + 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^k2), 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_{k1} and the largest of that (previous) row:
k  a(9*2^k2, ...) = 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^(i1))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

Cf. A160160, A139250, A139251, A160120, A160121, A160171, A170875.
Sequence in context: A197000 A198428 A083550 * A129279 A008496 A178573
Adjacent sequences: A160158 A160159 A160160 * A160162 A160163 A160164


KEYWORD

nonn,tabf,changed


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



