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A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4). 19
0, 1, 9, 21, 49, 61, 97, 133, 225, 237, 273, 309, 417, 453, 561, 669, 961, 973, 1009, 1045, 1153, 1189, 1297, 1405, 1729, 1765, 1873, 1981, 2305, 2413, 2737, 3061, 3969, 3981, 4017, 4053, 4161, 4197, 4305, 4413, 4737, 4773, 4881, 4989, 5313, 5421, 5745 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The structure has a fractal behavior similar to the toothpick sequence A139250.
First differences: A161415, where there is an explicit formula for the n-th term.
For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.
LINKS
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.]
FORMULA
a(n) = 1 + 4*A219954(n), n >= 1. - M. F. Hasler, Dec 02 2012
a(2^k) = (2^(k+1) - 1)^2. - Omar E. Pol, Jan 05 2013
EXAMPLE
From Omar E. Pol, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21, 49;
61, 97, 133, 225;
237, 273, 309, 417, 453, 561, 669, 961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A256530 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ |_| |
. |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|
. | |_| _ _ |_| |_| _ _ |_| |
. | | | _|_|_ _ _|_|_ | | |
. | _| |_| _ _ _ _ |_| |_ |
. | | |_ _| | _|_|_ | |_ _| | |
. | |_ _| | |_| _ |_| | |_ _| |
. | _ _ | _| |_| |_ | _ _ |
. | | _|_| | |_ _ _| | |_|_ | |
. | |_| _| |_ _| |_ _| |_ |_| |
. | | | |_ _ _ _ _ _ _| | | |
. | _| |_ _| |_ _| |_ _| |_ |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _| |_ _| |_ _| |_ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.
(End)
MAPLE
read("transforms") ; isA000079 := proc(n) if type(n, 'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc:
A160414 := proc(n) add( A161415(k), k=1..n) ; end proc: seq(A160414(n), n=0..90) ; # R. J. Mathar, Oct 16 2010
MATHEMATICA
A160414list[nmax_]:=Accumulate[Table[If[n<2, n, 4*3^DigitCount[n-1, 2, 1]-If[IntegerQ[Log2[n]], 2n, 0]], {n, 0, nmax}]]; A160414list[100] (* Paolo Xausa, Sep 01 2023, after R. J. Mathar *)
PROG
(PARI) my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<<valuation(n, 2)), n\2)); vector(99, i, 4*(s+=t(i))+1) \\ Altug Alkan, Sep 25 2015
CROSSREFS
Sequence in context: A259243 A241747 A133762 * A256530 A363634 A118130
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, May 20 2009
EXTENSIONS
Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009
More terms from R. J. Mathar, Oct 16 2010
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)