1,1

Limiting behavior of the rows of the triangle in A139251 when the first column of that triangle is omitted.

First differences of A159795. [From Omar E. Pol (info(AT)polprimos.com), Jul 24 2009]

David Applegate, Table of n, a(n) for n = 1..2047

Index entries for sequences related to toothpick sequences

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

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 = (n+1)2^n (where n is the row number), the second column

is 2^(n+1)+4, and the rest exhibit 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 inifitum.

Maple code from N. J. A. Sloane, May 18 2009:

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;

end;

Equals 2*A151688 and 4*A152980. [N. J. A. Sloane, Jul 16 2009]

Cf. A139250, A139251.

Cf. A153006, A159795. [From Omar E. Pol (info(AT)polprimos.com), Jul 24 2009]

Sequence in context: A120427 A060830 A080458 this_sequence A080229 A167523 A101887

Adjacent sequences: A147643 A147644 A147645 this_sequence A147647 A147648 A147649

nonn

David Applegate, Apr 30 2009