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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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17
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0, 1, 4, 25, 676, 458329, 210066388900, 44127887745906175987801, 1947270476915296449559703445493848930452791204, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352025
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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 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).
Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
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LINKS
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Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
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FORMULA
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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
a(n) is the number of root ancestral configurations for fully symmetric matching gene trees and species trees with 2^n leaves, a(n) = A355108(2^n). - Noah A Rosenberg, Jun 22 2022
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MATHEMATICA
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PROG
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(Haskell)
a004019 n = a004019_list !! n
a004019_list = iterate (a000290 . (+ 1)) 0
(Magma) [n le 1 select 0 else (Self(n-1)+1)^2: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015
(PARI) a(n) = if(n==0, 0, (a(n-1) + 1)^2);
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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