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A073121
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a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)) with a(1)=1 and (r,s)=(2,2).
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10
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1, 4, 10, 16, 28, 40, 52, 64, 88, 112, 136, 160, 184, 208, 232, 256, 304, 352, 400, 448, 496, 544, 592, 640, 688, 736, 784, 832, 880, 928, 976, 1024, 1120, 1216, 1312, 1408, 1504, 1600, 1696, 1792, 1888, 1984, 2080, 2176, 2272, 2368, 2464, 2560, 2656, 2752
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OFFSET
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1,2
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COMMENTS
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A recurrence occurring in the analysis of a regular expression algorithm.
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REFERENCES
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K. Ellul, J. Shallit and M.-w. Wang, Regular expressions: new results and open problems, in Descriptional Complexity of Formal Systems (DCFS), Proceedings of workshop, London, Ontario, Canada, 21-24 August 2002, pp. 17-34.
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LINKS
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Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu, Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012-2014.
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FORMULA
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a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c.
a(n) == 1 (mod 3), a(n+1)-a(n) = 3*A053644(n). If k >= 1: a(2^k)=4^k, a(3*2^k)=(10/9)*4^k. More generally a(m*2^k) = a(m)*4^k. Hence for any n, n^2 <= a(n) <= C*n^2 where C is a constant 1.125 < C < 1.14 and it seems that C = lim_{k->infinity} a(A001045(k))/A001045(k)^2 where A001045(k) =(2^n - (-1)^n)/3 is the Jacobsthal sequence. In other words, in the range 2^k <= n <= 2^(k+1) the maximum of a(n)/n^2 is reached for the only possible n in the Jacobsthal sequence. - Benoit Cloitre, Aug 26 2002
a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c
G.f.: 3*x/(1-x)^2 * ((2*x+1)/3 + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003
G.f.: A(x) = 2 * (1/x + 2 + x) * A(x^2) - x. - Michael Somos, Jul 04 2017
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EXAMPLE
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a(1)=1, a(2) = 2*(a(1)+a(1)) = 4, a(3) = 2*(a(2)+a(1)) = 10.
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MAPLE
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a:= proc(n) option remember; `if`(n=1, 1,
2*((m-> a(m)+a(n-m))(iquo(n, 2))))
end:
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Module[{m = 1, A = 1}, While[m < n, m *= 2; A = (Normal[A] /. x -> x^2) 2 (1 + x)^2 - 1 + O[x]^m]; Coefficient[A, x, n - 1]]]; (* Michael Somos, Jul 04 2017 *)
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PROG
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(Haskell)
a073121 n = a053644 n * (fromIntegral n + 2 * a053645 n)
(PARI) {a(n) = n--; if( n<0, 0, my(m=1, A = 1 + O(x)); while(m<=n, m*=2; A = subst(A, x, x^2) * 2 * (1 + x)^2 - 1); polcoeff(A, n))}; /* Michael Somos, Jul 04 2017 */
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CROSSREFS
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Sequences of form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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