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A088720 Unique monotone sequence satisfying a(a(a(n))) = 2n. 3
4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

For k >= 1 and m >= 2, a monotone a(n) such that a^(k+1)(n) = mn is unique only when m = 2 or (k,m) = (1,3).

REFERENCES

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

LINKS

Robert Israel, Table of n, a(n) for n = 3..10000

FORMULA

For a^(k+1)(n) = 2n, we have for (k+1)2^m <= n <= (2k+1)2^m, a(n) = n+2^m; for (2k+1)2^m <= n <= (2k+2)2^m, a(n) = 2n-2k.2^m.

From Robert Israel, Apr 05 2017: (Start)

a(2n) = 2*a(n).

a(4n+1) = a(2n+1)+2*a(n).

a(4n+3) = 3*a(2n+1)-2*a(n).

G.f. g(z) satisfies g(z) = 4*z^3+5*z^4+2*z^5-3*z^7+5*z^9-4*z^11+(2+1/(2*z)+3*z/2)*g(z^2)-(1/(2*z)+3*z/2)*g(-z^2)+(2*z-2*z^3)*g(z^4).

(End)

EXAMPLE

a(a(a(3))) = a(a(4)) = a(5) = 6.

MAPLE

seq(op([seq(n+2^m, n=3*2^m .. 5*2^m-1), seq(2*n-4*2^m, n=5*2^m..6*2^m-1)]), m=0..10); # Robert Israel, Apr 05 2017

CROSSREFS

Cf. A007378, A003605, A088721.

Sequence in context: A257816 A033597 A336379 * A134848 A061578 A284902

Adjacent sequences:  A088717 A088718 A088719 * A088721 A088722 A088723

KEYWORD

easy,nonn

AUTHOR

Colin Mallows, Oct 16 2003

EXTENSIONS

More terms from John W. Layman, Oct 18 2003

STATUS

approved

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Last modified June 21 18:50 EDT 2021. Contains 345365 sequences. (Running on oeis4.)