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A193494
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Worst case of an unbalanced recursive algorithm over all n-node binary trees.
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1
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0, 1, 2, 4, 5, 7, 9, 13, 14, 16, 18, 22, 24, 28, 32, 40, 41, 43, 45, 49, 51, 55, 59, 67, 69, 73, 77, 85, 89, 97, 105, 121, 122, 124, 126, 130, 132, 136, 140, 148, 150, 154, 158, 166, 170, 178, 186, 202, 204, 208, 212, 220, 224, 232, 240, 256, 260
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OFFSET
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0,3
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COMMENTS
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Solves the recurrence a(0) = 0, a(n+1) = 1 + Max_{0 <= k <= n-k} (2*a(k) + a(n-k)).
a(floor(n/2)) is also the number of white areas in the elementary cellular automata based on rule 126 completed up to generation n. - Philippe Beaudoin, Feb 03 2014
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REFERENCES
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(I think I've seen it years ago, but I have no idea where.)
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LINKS
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Eric Weisstein's World of Mathematics, Rule 126.
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FORMULA
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a(n) = O(n^(log_2(3)));
a(n) = 2^(nu(n)-1) + a(n-1) when n>0 and has nu(n) 1-bits in binary (A000120);
If n = 2^(e_1) + ... + 2^(e_t) with e_1 > ... > e_t >= 0, then a(n) = ((3^(e_1) + 2*3^(e_2) + ... + 2^(t-1)*3^(e_t) + 2^t-1)/2;
The generating function has the form (t_0 + t_0*t_1 + t_0*t_1*t_2 + ...)/ (1-z), where t_0 = z and t_k = z^{2^{k-1}} + 2*z^{2^k} for k > 0.
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MAPLE
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a:= proc(n) option remember;
`if`(n=0, 0, 1 +max(seq(2*a(k)+a(n-1-k), k=0..(n-1)/2)))
end:
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MATHEMATICA
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a[0]=0; a[n_]:=a[n]=1+Max[Table[2a[k]+a[n-1-k], {k, 0, (n-1)/2}]]
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PROG
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(PARI) a=vector(60); a[1]=0; for(n=0, #a-2, a[n+2]=1+vecmax(vector(n\2+1, k, 2*a[k]+a[n-k+2]))); a \\ Charles R Greathouse IV, Jul 29 2011
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CROSSREFS
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log_2(a(n) - a(n-1)) is A000120(n) - 1, for n > 0.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Third formula corrected by Don Knuth, Dec 08 2011
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STATUS
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approved
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