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%I A147562
%S A147562 0,1,5,9,21,25,37,49,85,89,101,113,149,161,197,233,341,345,357,369,405,
417,
%T A147562 453,489,597,609,645,681,789,825,933,1041,1365,1369,1381,1393,1429,1441,
%U A147562 1477,1513,1621,1633,1669,1705,1813,1849,1957,2065,2389,2401,2437,2473
%N A147562 Number of "ON" cells at nth stage in simple 2-dimensional cellular automaton
(see Comments for precise definition).
%C A147562 Studied by Holladay and Ulam circa 1960. See Fig. 1 and Example 1 of
the Ulam reference. - N. J. A. Sloane, Aug 02 2009.
%C A147562 Singmaster calls this the Ulam-Warburton CA. - N. J. A. Sloane, Aug 05
2009
%C A147562 On the infinite square grid, start with all cells OFF.
%C A147562 Turn a single cell to the ON state.
%C A147562 At each subsequent step, each cell with exactly one neighbor ON is turned
ON, and everything that is already ON remains ON.
%C A147562 Here "neighbor" refers to the four adjacent cells in the X and Y directions.
%C A147562 Note that "neighbor" could equally well refer to the four adjacent cells
in the diagonal directions, since the graph formed by Z^2 with "one-step
rook" adjacencies is isomorphic to Z^2 with "one-step bishop" adjacencies.
%D A147562 D. Singmaster, On the cellular automaton of Ulam and Warburton, unpublished
manuscript, 2003.
%D A147562 S. Ulam, On some mathematical problems connected with patterns of growth
of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems
in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14,
Amer. Math. Soc., 1962.
%H A147562 David Applegate,
The movie version
%H A147562 Index entries for sequences related to cellular
automata
%H A147562 O. E. Pol,
Illustration of initial terms (One-step rook) [From Omar E. Pol
(info(AT)polprimos.com), Nov 02 2009]
%H A147562 O. E. Pol,
Illustration of initial terms (One-step bishop) [From Omar E.
Pol (info(AT)polprimos.com), Nov 02 2009]
%H A147562 O. E. Pol,
Illustration of initial terms (Overlapping squares) [From Omar
E. Pol (info(AT)polprimos.com), Nov 02 2009]
%H A147562 O. E. Pol,
Illustration of initial terms (Overlapping X-toothpicks) [From
Omar E. Pol (info(AT)polprimos.com), Nov 04 2009]
%H A147562 O. E. Pol,
Illustration of initial terms of A139250, A160120, A147562(Overlapping
figures) [From Omar E. Pol (info(AT)polprimos.com), Nov 04 2009]
%F A147562 For n>0, a(n) = 1 + 4*Sum_{k=1..n} 3^(wt(k-1)-1), where wt() = A000120().
%e A147562 If we label the generations of cells turned ON by consecutive numbers
we get a rosetta cell pattern (1 should be added to these numbers
to match the new offset):
%e A147562 ................3
%e A147562 ...............323
%e A147562 ..............3.1.3
%e A147562 .............3210123
%e A147562 ..............3.1.3
%e A147562 ...............323
%e A147562 ................3
%e A147562 In the first generation, only the central "0" is ON, a(1)=1. In the next
generation, we turn ON four "1", leading to a(2)=a(1)+4=5. In the
third generation, four "2" are turned ON, a(3)=a(2)+4=9. In the fourth
generation, each of the four wings allows three 3's to be turned
ON, a(4)=a(3)+4*3=21.
%p A147562 # [x,y] coordinates of cells on
%p A147562 Lse := [[0,0]] ;
%p A147562 # enclosing rectangle of the cells on (that is, minima and maxima in
Lse)
%p A147562 xmin := 0 ;
%p A147562 xmax := 0 ;
%p A147562 ymin := 0 ;
%p A147562 ymax := 0 ;
%p A147562 # count neighbors of x,y which are on; return 0 if [x,y] is in L
%p A147562 cntnei := proc(x,y,L)
%p A147562 local a,p,xpt,ypt;
%p A147562 a := 0 ;
%p A147562 if not [x,y] in L then
%p A147562 for p in Lse do
%p A147562 xpt := op(1,p) ;
%p A147562 ypt := op(2,p) ;
%p A147562 if ( abs(xpt-x) = 1 and ypt=y ) or ( x=xpt and abs(ypt-y) = 1) then
%p A147562 a := a+1 ;
%p A147562 fi;
%p A147562 od:
%p A147562 fi:
%p A147562 RETURN(a) ;
%p A147562 end:
%p A147562 # loop over generations/steps
%p A147562 for stp from 1 to 10 do
%p A147562 Lnew := [] ;
%p A147562 for x from xmin-1 to xmax+1 do
%p A147562 for y from ymin-1 to ymax+1 do
%p A147562 if cntnei(x,y,Lse) = 1 then
%p A147562 Lnew := [op(Lnew),[x,y]] ;
%p A147562 fi;
%p A147562 od:
%p A147562 od:
%p A147562 for p in Lnew do
%p A147562 xpt := op(1,p) ;
%p A147562 ypt := op(2,p) ;
%p A147562 xmin := min(xmin,xpt) ;
%p A147562 xmax := max(xmax,xpt) ;
%p A147562 ymin := min(ymin,ypt) ;
%p A147562 ymax := max(ymax,ypt) ;
%p A147562 od:
%p A147562 Lse := [op(Lse),op(Lnew)] ;
%p A147562 print(nops(Lse)) ;
%t A147562 Map[Function[Apply[Plus,Flatten[ #1]]],CellularAutomaton[{686,{2,{{0,
2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},200]] (Nadia Heninger and
N. J. A. Sloane, Aug 11 2009)
%Y A147562 Cf. A000120, A139250, A147582 (number turned on at n-th step), A147610.
%Y A147562 Cf. A130665, A151920, A160120, A160410, A160414. [From Omar E. Pol (info(AT)polprimos.com),
Nov 02 2009]
%Y A147562 Sequence in context: A092387 A160720 A147552 this_sequence A162795 A147407
A146867
%Y A147562 Adjacent sequences: A147559 A147560 A147561 this_sequence A147563 A147564
A147565
%K A147562 nonn,nice,new
%O A147562 0,3
%A A147562 N. J. A. Sloane (njas(AT)research.att.com), based on emails from Franklin
T. Adams-Watters, Richard Mathar and David Wilson, Apr 29 2009
%E A147562 Offset and initial terms changed by N. J. A. Sloane, Jun 07 2009
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