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A147646
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If A139251 is written as a triangle with rows of lengths 1, 2, 4, 8, 16, ..., the n-th row begins with 2^n followed by the first 2^n-1 terms of the present sequence.
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13
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4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 36, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 192, 68, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 196, 88, 60, 92, 112, 140, 208, 260, 224, 172, 212, 296, 364, 488, 672, 704, 448, 132
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OFFSET
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1,1
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COMMENTS
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Limiting behavior of the rows of the triangle in A139251 when the first column of that triangle is omitted.
It appears that a(n) is also the number of new grid points that are covered at n-th stage of A139250 version "Tree", assuming the toothpicks have length 4, 3, and 2 (see also A159795 and A153006). - Omar E. Pol, Oct 25 2011
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LINKS
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FORMULA
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Letting n = 2^i + j for 0 <= j < 2^i, we have the recurrence (see A139251 for proof):
a(1) = 4
a(2) = 8
a(n) = 2n+4 = 2*a(n/2) - 4 if j = 0
a(n) = 2*a(j) + a(j+1) - 4 if j = 2^i-1
a(n) = 2*a(j) + a(j+1) if 1 <= j < 2^i-1
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EXAMPLE
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Further comments: A139251 as a triangle is:
. 1
. 2 4
. 4 4 8 12
. 8 4 8 12 12 16 28 32
. 16 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80
. 32 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...
leading to the present sequence:
. 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...
Note that this can also be written as a triangle:
. 4 8
. 12 12 16 28
. 32 20 16 28 36 40 60 88
. 80 36 16 28 36 40 60 88 84 56 60 92 112 140 208 256
. 192 68 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
The first column is (n+1)2^n (where n is the row number),
the second column is 2^(n+1)+4,
and the rest exhibits the same constant column behavior,
where the rows converge to:
. 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
Once again this can be written as a triangle:
. 16
. 28 36 40 60
. 88 84 56 60 92 112 140 208
. 256 196 88 60 92 112 140 208 260 224 172 212 296 364 488 672
. 704 452 152 60 92 112 140 208 260 224 172 212 296 364 488 672 708 480 236 ...
and this behavior continues ad infinitum.
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MAPLE
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S:=proc(n) option remember; local i, j;
if n <= 0 then RETURN(0); fi;
if n <= 2 then RETURN(2^(n+1)); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then RETURN(2*n+4); fi;
if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i-1 then RETURN(2*S(j)+S(j+1)-4); fi;
-1;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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