

A063882


a(n) = a(n  a(n  1)) + a(n  a(n  4)), with a(1) = ... = a(4) = 1.


31



1, 1, 1, 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 39, 40
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OFFSET

1,5


COMMENTS

A captivating recursive function. A metaFibonacci recursion.
This has been completely analyzed by Balamohan et al. They prove that the sequence a(n) is monotonic, with successive terms increasing by 0 or 1, so the sequence hits every positive integer.
They demonstrate certain special structural properties and periodicities of the associated frequency sequence (the number of times a(n) hits each positive integer) that make possible an iterative computation of a(n) for any value of n.
Further, they derive a natural partition of the asequence into blocks of consecutive terms ("generations") with the property that terms in one block determine the terms in the next.


LINKS

A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 11281147. (20 pages); DOI:10.1137/15M1040505


FORMULA

n/2 < a(n) <= n/2 + log_2 (n)  1 for all n > 6 [Balamohan et al., Proposition 5].


MAPLE

a := proc(n) option remember; if n<=4 then 1 else if n > a(n1) and n > a(n4) then RETURN(a(na(n1))+a(na(n4))); else ERROR(" died at n= ", n); fi; fi; end;


MATHEMATICA

a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = a[na[n1]] + a[na[n4]]


PROG

(Haskell)
a063882 n = a063882_list !! (n1)
a063882_list = 1 : 1 : 1 : 1 : zipWith (+)
(map a063882 $ zipWith () [5..] a063882_list)
(map a063882 $ zipWith () [5..] $ drop 3 a063882_list)


CROSSREFS



KEYWORD

nice,nonn


AUTHOR

Theodor Schlickmann (Theodor.Schlickmann(AT)cec.eu.int), Aug 28 2001


EXTENSIONS



STATUS

approved



