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%I #36 Feb 24 2021 02:48:18
%S 0,1,2,4,8,8,8,8,16,32,56,32,16,8,16,32,56,56,64,80,152,232,352,144,
%T 48,32,24,40,56,56,64,80,152,232,352,216,168,176,272,360,496,448,536,
%U 664,1168,1488,2000,768,304,336,264,192,112,120,128,112,168,240,352,216,168,176,272,360,496
%N First differences of the 3D toothpick numbers A160160.
%C Number of toothpicks added at n-th stage to the three-dimensional toothpick structure of A160160.
%C 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
%H M. F. Hasler, <a href="/A160161/b160161.txt">Table of n, a(n) for n = 0..500</a>
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F 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
%e Array begins:
%e ===================
%e x y z
%e ===================
%e 0 1
%e 2 4 8
%e 8 8 8
%e 16 32 56
%e 32 16 8
%e 16 32 56
%e 56 64 80
%e 152 232 352
%e 144 48 32
%e ...
%e From _Omar E. Pol_, Feb 28 2018: (Start)
%e 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:
%e 1, 2, 4;
%e 8, 8, 8;
%e 8, 16, 32, 56, 32, 16;
%e 8, 16, 32, 56, 56, 64, 80, 152, 232, 352, 144, 48;
%e 32, 24, 40, 56, 56, 64, 80, 152, 232, 352, 216, 168, 176, 272, 360, 496, 448, ...
%e (End)
%e 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:
%e k | a(9*2^k-2, ...) = A_k ; B_k ; A_k
%e ---+-------------------------------------
%e | a( 1 .. 6) = (1, 2, 4, 8, 8, 8) (One might consider a row (8 ; 8 ; 8).)
%e 0 | a( 7, ...) = (8, 16, 32 ; 56, 32, 16 ; 8, 16, 32)
%e 1 | a(16, ...) = (56, 56, 64, 80, 152, 232 ; 352, 144, 48, 32, 24, 40 ;
%e | 56, 56, 64, 80, 152, 232)
%e 2 | a(34, ...) = (352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488 ;
%e | 2000, 768, 304, 336, 264, 192, 112, 120, 128, 112, 168, 240 ;
%e | 352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488)
%e 3 | a(70, ...) = (2000, 984, ... ; 10576, 4304, ... ; 2000, 984, ...)
%e 4 | a(142, ...) = (10576, 5016, ... ; 54328, 24120, ...; 10576, 5016, ...)
%e etc. - _M. F. Hasler_, Dec 11 2018
%o (PARI) A160161_vec(n)=(n=A160160_vec(n))-concat(0,n[^-1]) \\ _M. F. Hasler_, Dec 11 2018
%o (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
%Y Cf. A160160, A139250, A139251, A160120, A160121, A160171, A170875.
%K nonn,tabf
%O 0,3
%A _Omar E. Pol_, May 03 2009
%E Extended to 78 terms with C++ program by _R. J. Mathar_, Jan 09 2010
%E Edited and extended by _M. F. Hasler_, Dec 11 2018
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