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A169689
(A169648(4n+4) - A147582(4n+5))/4.
4
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
OFFSET
-1,3
COMMENTS
A169648 and A147582 agree except at these terms.
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(-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.
EXAMPLE
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,
...
MAPLE
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)];
CROSSREFS
Equals A169688/4. Cf. A169697 (limit of rows).
Sequence in context: A318209 A120462 A236602 * A328757 A061592 A081631
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 14 2010
STATUS
approved