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A004019
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a(0) = 0; for n>0, a(n) = (a(n-1) + 1)^2.
(Formerly M3611)
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10
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0, 1, 4, 25, 676, 458329, 210066388900, 44127887745906175987801, 1947270476915296449559703445493848930452791204, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352025
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Take the standard rooted binary tree of depth n, with 2^(n+1) - 1 labeled nodes. Here is a poor picture of the tree of depth 3:
.......R
...../...\
..../.....\
...o.......o
../.\...../.\
.o...o...o...o
/.\./.\./.\./.\
o o o o o o o o
Let the number of rooted subtrees be s(n). For example, for n = 1 the s(2) = 4 subtrees are:
R...R...R......R
.../.....\..../.\
..o.......o..o...o
Then s(n+1) = 1 + 2*s(n) + s(n)^2 = (1+s(n))^2 and so s(n) = a(n+1).
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Elmar Teufl and Stephan Wagner, Enumeration problems for classes of self-similar graphs, Journal of Combinatorial Theory, Series A, Volume 114, Issue 7, October 2007, Pages 1254-1277.
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..15
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index entries for sequences related to rooted trees
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FORMULA
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a(n) = A003095(n)^2 = A003095(n+1) - 1 = A056207(n+1) + 1.
It follows from Aho and Sloane that there is a constant c such that a(n) is the nearest integer to c^(2^n). In fact a(n+1) = nearest integer to b^(2^n) - 1 where b = 2.25851845058946539883779624006373187243427469718511465966.... - Henry Bottomley, Aug 30 2005.
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MATHEMATICA
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Table[Nest[(1 + #)^2 &, 0, n], {n, 0, 12}] (* From Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
NestList[(#+1)^2&, 0, 10] (* From Harvey P. Dale, Oct 08 2011 *)
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PROG
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(Haskell)
a004019 n = a004019_list !! n
a004019_list = iterate (a000290 . (+ 1)) 0
-- Reinhard Zumkeller, Feb 01 2013
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CROSSREFS
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Cf. A001699, A056207.
Cf. A000290.
Sequence in context: A167041 A123129 A075577 * A072882 A014253 A132553
Adjacent sequences: A004016 A004017 A004018 * A004020 A004021 A004022
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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One more term from Henry Bottomley, Jul 24 2000
Additional comments from Max Alekseyev, Aug 30 2005
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STATUS
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approved
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