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0, 1, 6, 4, 24, 4, 20, 12, 84, 4, 20, 12, 76, 12, 60, 36, 276, 4, 20, 12, 76, 12, 60, 36, 260, 12, 60, 36, 228, 36, 180, 108, 876, 4, 20, 12, 76, 12, 60, 36, 260, 12, 60, 36, 228, 36, 180, 108, 844, 12, 60, 36, 228, 36, 180, 108, 780, 36, 180, 108, 684, 108, 540, 324, 2724, 4
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OFFSET
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-1,3
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COMMENTS
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A169648 and A147582 agree except at these terms.
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LINKS
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Table of n, a(n) for n=-1..64.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
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FORMULA
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a(-1)=0, a(0)=1, a(1)=6. For n >= 2, let n = 2^k+j with 0 <= j < 2^k, and write j+1 = 2^m*(2t+1). Then a(n) = 4*(3^(m+1)-2^(m+1))*3^wt(t), except if j=2^k-1 we must add 2^(k+1) to the result (here wt(t) = A000120(t)).
Recurrence: a(-1)=0, a(0)=1, a(1)=6. For n>=2, write n = 2^k + j, with 0 <= j < 2^k. If j+1 is a power of 2, say j+1 = 2^r, set f=j+1 if r<k, f=3(j+1) if r=k, and otherwise set f=0. Then a(n) = 3*a(j) + f. (The explicit formula in the previous line is better.)
Since there is a simple explicit formula for A147582(n), this provides a simple way to generate A169648.
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EXAMPLE
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Can be written in the form of a triangle:
0,
1,
6,
4,24,
4,20,12,84,
4,20,12,76,12,60,36,276,
4,20,12,76,12,60,36,260,12,60,36,228,36,180,108,876,
4,20,12,76,12,60,36,260,12,60,36,228,36,180,108,844,12,60,36,228,36,180,108,780,36,180,108,684,108,540,324,2724,
...
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MAPLE
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a:=proc(n) option remember; local f, j, k, t1;
if n=-1 then RETURN(0); elif n=0 then RETURN(1); elif n=1 then RETURN(6);
else k:=floor(log(n)/log(2)); j:=n-2^k; t1 := 2^floor(log(j+1)/log(2));
if t1=j+1 and j < 2^k-1 then f := j+1 elif t1=j+1 then f := 3*(j+1) else f := 0; fi;
RETURN(3*a(j)+f);
fi;
end;
[seq(a(n), n=-1..200)];
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CROSSREFS
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Equals A169688/4. Cf. A169697 (limit of rows).
Sequence in context: A213573 A185734 A120462 * A061592 A081631 A137174
Adjacent sequences: A169686 A169687 A169688 * A169690 A169691 A169692
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KEYWORD
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nonn,tabf
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AUTHOR
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N. J. A. Sloane, Apr 14 2010
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STATUS
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approved
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